STRAIGHT LINE COORDINATE GEOMETRY

Save this PDF as:

Size: px
Start display at page:

Transcription

1 STRAIGHT LINE COORDINATE GEOMETRY (EXAM QUESTIONS)

2 Question 1 (**) The points P and Q have coordinates ( 7,3 ) and ( 5,0), respectively. a) Determine an equation for the straight line PQ, giving the answer in the form ax + by + c = 0, where a, b and c are integers. The straight line RT with equation 3x + 5y = 19 intersects the straight line PQ at the point R. b) Find the coordinates of R. x 4y + 5 = 0, R ( 3,2)

3 Question 2 (**) The straight line 1 l passes through the points A( 1,3) and ( 2, 3) B. a) Find an equation for l 1. The straight line l 2 has equation 2y = x + 3. b) Find the exact coordinates of the point of intersection between l 1 and l 2. y = 1 2x, ( 1, ) Question 3 (**) The coordinates of three points are given below A( 2,1), B( 6, 1) and ( 5,2) C. Determine the equation of the straight line which passes through C and is parallel to AB, giving the answer in the form ax + by = c, where a, b and c are integers. x + 4y = 13

4 Question 4 (**) The straight line l passes through the points with coordinates ( 4,9) and ( 4, 3). a) Find an equation for l, giving the answer in the form ax + by = c where a, b and c are integers. The straight line l meets the coordinate axes at the points A and B. b) Determine the area of the triangle OAB, where O is the origin. 3x + 2y = 6, area = 3

5 Question 5 (**) The straight line 1 l passes through the points A ( 5,4) and ( 13,0) B. a) Find an equation of l 1, in the form ax + by = c, where a, b and c are integers. The straight line 2 l passes through the point ( 0,2) C and has gradient 4. b) Write down an equation of l 2. The point P is the intersection of l 1 and l 2. c) Determine the exact coordinates of P. x + 2y = 13, y 2 4x =, P 9 50 (, ) 7 7

6 Question 6 (**) The straight line 1 l passes through the points A ( 3,17) and ( 13, 3) B. a) Find an equation of l 1. The straight line l 2 has gradient 1 3 and passes through the point C ( 0,2 ). b) Find an equation of l 2. The two lines, l 1 and l 2, intersect at the point D. c) Show that the length of AD is k 5, where k is an integer. y + 2x = 23, 3y = x + 6, k = 6

7 Question 7 (**) The straight line l has a gradient of 5 12 B( k,11), where k is a constant., and passes through the points ( 10,1) A and a) Find an equation of l, in the form ax + by = c, where a, b and c are integers. b) Determine the value of k. c) Hence show that the distance AB is 26 units. 5x + 12y = 62, k = 14

8 Question 8 (**) The points A and B have coordinates ( 1,1 ) and ( 5,7 ), respectively. a) Find an equation for the straight line l 1 which passes through A and B. The straight line l 2 with equation 2x + 3y = 18 meets l 1 at the point C. b) Determine the coordinates of C. The point D, where x = 3, lies on l 2. c) Show clearly that AD = BD. 2y 3x 1 =, C ( 3,4)

9 Question 9 (**+) The straight line L passes through the points ( 2,5 ) and ( 2,3) coordinate axes at the points P and Q., and meets the Find the area of a square whose side is PQ. MP1-O, area = 80

10 Question 10 (**+) The straight line 1 constant. l passes through the points A( 1, 1) and B( k,5), where k is a a) Given that the gradient of l 1 is 1 2 show that k = 11. The straight line l 2 passes through the midpoint of AB and is perpendicular to l 1. b) Determine an equation of l 2, giving the answer in the form ax + by = c, where a, b and c are integers. c) Calculate the area of the triangle enclosed by l 2 and the coordinate axes. 2x + y = 12, area = 36

11 Question 11 (**+) The straight lines l 1 and l 2 have equations l 1 : 2x + y = 10, l 2 : 3x 4y = 10. a) Sketch l 1 and l 2 in a single set of axes. The sketch must include the coordinates of all the points where each of these straight lines meet the coordinate axes. The two lines intersect at the point P. b) Use algebra to determine the exact coordinates of P. P 50 (, 10 ) 11 11

12 Question 12 (**+) The points A and B have coordinates ( 1,4 ) and ( 3, 2) B, respectively. a) Find an equation of the straight line L which is perpendicular to the straight line AB and passes through the point B, giving the answer in the form ax + by + c = 0 where a, b and c are integers. L meets the coordinate axes at the points C and D. The point O represents the origin. b) Find the area of the triangle OCD. 2x 3y 12 = 0, area = 12

13 Question 13 (**+) The straight line l 1 has equation 3x 2y = 1. a) Find an equation of the straight line l 2 which is perpendicular to l 1 and passes through the point A( 4, 1), giving the answer in the form ax + by = c where a, b and c are integers. The straight line l 1 meets the coordinate axes at the points P and Q. The point O represents the origin. b) Show that the area of the triangle OPQ is 1 12 of a square unit. C1A, 2x + 3y = 5

14 Question 14 (**+) The straight line 1 L passes through the points A( 6,4) and ( 3,16) B. a) Find an equation for L 1. The straight line 2 L passes through the points C ( 9, 1) and ( 7,11) D. b) Find an equation for L 2. c) Show that L 1 is perpendicular to L 2 The point E is the of intersection of L 1 and L 2. d) Show that the coordinates of E are ( 3,8). 3y 4x = 36, 4y + 3x = 23

15 Question 15 (**+) y B L 3 C A O L 2 x L 1 The figure above shows three straight lines L 1, L 2 and L 3. a) Find an equation of the straight line L 1, given that it passes through the points A ( 7,3) and ( 9,9) B. L 2 is perpendicular to L 1 and passes through A. b) Find an equation of L 2. L 3 meets L 1 at the point B and L 2 at the point C. The equation of L 3 is x + 9 y =. 2 c) Determine the coordinates of C. d) Show that the triangle ABC is isosceles. C1G, y = 3x 18, 3y x 16 + =, C ( 1,5 )

16 Question 16 (**+) The straight line L passes through the points A( 1, 14) and ( 3, 2) B. a) Find an equation for L, giving the answer in the form y = mx + c. The point C has coordinates ( 100, 312). b) Determine, by calculation, whether C lies above L or below L. C1I, y = 3x 11, below

17 Question 17 (**+) The straight line 1 l passes through the points A( 1, 1) and ( 7,8) B. a) Find an equation of l 1. b) Show that the length of AB is k 13, where k is an integer. The straight line l 2, whose equation is x + y = 10, meets l 1 at the point C. The point D is the y intercept of l 2. c) Determine the area of the triangle OCD, where O is the origin. 2y = 3x 5, k = 3, area = 25

18 Question 18 (**+) The points A and B have coordinates ( 2,3 ) and ( 6,1 ), respectively. a) Find an equation of the straight line l 1 which passes through A and B. The line 2 l has gradient 1 4 and meets the y axis at the point 13 ( ) 0, 4. The two lines, l 1 and l 2, intersect at the point C. b) Show clearly that the length of AC is exactly y + x = 8 Question 19 (***) The straight line l passes through the points A ( 4,3) and ( 4,9) B. a) Find an equation for l. l meets the coordinate axes at the points C and D. b) Show that the midpoint of CD is at a distance of 5 units from the origin. 4y + 3x = 24

19 Question 20 (***) A triangle has vertices A ( 2,6), B( 2,8) and ( 1,0 ) C. a) Show that the triangle is right angled. b) Calculate the area of the triangle. area = 15 Question 21 (***) The straight line l 1 has gradient 2 and passes through the point ( 8,3 ). a) Find an equation for l 1. The straight line l 2 is perpendicular to the line with equation 3x y + 10 = 0 and crosses the y axis at ( 0,1 ). b) Determine an equation for l 2, giving the answer in the form ax + by = c, where a, b and c are integers. c) Determine the coordinates of the point of intersection between l 1 and l 2. y = 2x 13, x 3y 3 + =, ( 6, 1)

20 Question 22 (***) The points A and B have coordinates ( 3, 5) and B ( 1,1), respectively. a) Find an equation of the straight line through A and B, in the form ax + by = c where a, b and c are integers. The midpoint of AB is M, and the line segment MC is perpendicular to AB. b) Find the coordinates of M. c) State the gradient of MC. d) Given that C has coordinates ( 8,a ), find the value of a. 3x y 4 M, m = 1, a = =, ( 2, 2)

21 Question 23 (***) The straight line L 1 passes through the point ( 1,2 ) joins the points P ( 7,4) and Q ( 3,8). A and is parallel to the line that a) Show that an equation for L 1 is x + y = 1. The straight line L 2 is perpendicular to 1 intersecting L 1 at the point C. L and passes through the point ( 4,3) B, b) Find an equation for L 2. c) Determine the coordinates of C. y = x + 7, C ( 3,4)

22 Question 24 (***) The straight line 1 L passes through the points A ( 13,5) and ( 9,2) B. a) Find an equation for L 1. The point D lies on L 1 and the point C has coordinates ( 2,3 ). The straight line L 2 passes through C and D, and is perpendicular to L 1. b) Determine an equation for L 2, giving the answer in the form ax + by = c, where a, b and c are integers. c) Find the coordinates of D. 4y = 3x 19, 3y 4x 17 + =, D( 5, 1)

23 Question 25 (***) A( 4,3) B C ( 8, 7) D The figure above shows a kite ABCD, where the vertices A and C have coordinates 8, 7, respectively. ( 4,3 ) and ( ) The diagonal BD is a line of symmetry of the kite. Find an equation for the diagonal BD. MP1-N, 5y = 2x 22

24 Question 26 (***) The points A ( 0,3), B( 2, 1) and (,1) C k are given, where k is a constant. a) Find the exact length of AB. b) Given that AB is perpendicular to BC, find the value of k. c) Determine the area of the triangle ABC. AB = 20 = 2 5, k = 6, area = 10

25 Question 27 (***) y R Q P O x The figure above shows the right angled triangle PQR, whose vertices are located at P( 6, 2) and ( 6,6) Q. It is further given that PQR = 90. a) Find an equation for the straight line l, which passes through Q and R. The straight line l meets the x axis at the point T. b) Given that Q is the midpoint of RT, determine the coordinates of R. C1J, 2y 3x 30 + =, R ( 2,12)

26 Question 28 (***) x 2 = 4y + 12 y Q 2y 3x = 2 P A x The figure above shows the curve C and the straight line L, with respective equations x 2 = 4y + 12 and 2y 3x = 2. C meets the y axis at the point A, while C and L intersect each other at the points P and Q. a) Find the coordinates of P and the coordinates of Q. b) Show clearly that PAQ = 90. C1M, P( 2, 2), Q ( 8,13)

27 Question 29 (***). y A E B l O D x C not drawn accurately The figure above shows a triangle with vertices at A ( 2,6), B ( 11,6 ) and (, ) C p q. a) Given that the point D ( 6,2) is the midpoint of AC, determine the value of p and the value of q. The straight line l, passes through D and is perpendicular to AC. The point E is the intersection of l and AB. b) Find the coordinates of E. C1Q, p = 10, 2 q =, E ( 10,6)

28 Question 30 (***) The straight line PQ has equation 5x 2y + 7 = 0. The points P and Q have coordinates ( 1,1) and ( 1,k ), respectively. Calculate, showing a clear method a) the gradient of PQ. b) the value of k. c) the distance PR, where R is the point ( 8,0). gradient = 5, k = 6, PR = 5 2 2

29 Question 31 (***) The points A and B have coordinates ( 1,5 ) and ( 7,11 ), respectively. a) Show that the equation of the perpendicular bisector of AB is 4x + 3y = 36. The perpendicular bisector of AB crosses the coordinate axes at the points P and Q. b) Find the area of the triangle OPQ, where O is the origin. SYN-A, area = 54

30 Question 32 (***) The points A, B and C have coordinates ( 1,5 ), ( 7, 1) and ( ) 6,6, respectively. a) Find an equation of the straight line through A and B, giving the answer in the form ax + by = c, where a, b and c are integers. The midpoint of AB is the point M. b) Find the coordinates of M. c) Show that MC is perpendicular to AB. d) Calculate the area of the triangle ABC. 3x 4y 17 + =, ( 3,2) M, area = 25

31 Question 33 (***) The points A, B, C and D have coordinates ( 5,6), ( 5,1 ), ( 8,3 ) and (, 13) respectively, where k is a constant. a) Find an equation of the straight line through A and B. b) Given that CD is perpendicular to AB, find the value of k. k, x + 2y = 7, k = 0

32 Question 34 (***) Relative to a fixed origin O the points A, B and C have respective coordinates 1,3 13,k, where k is a constant. ( ), ( 1,11 ) and ( ) a) Find the length of AB, in the form a 17, where a is an integer. b) Given the length of BC is 3 17, determine the possible values of k. The actual value of k is in fact the smaller of the two values found in part (b). c) Show clearly that ABC = 90. d) Calculate the area of the triangle ABC. AB = 2 17, k = 8 or 14, area = 51

33 Question 35 (***) L 2 y L 1 P Q O R x The figure above shows the straight line L 1 with equation The line L 2 is perpendicular to 1 3x 2y + 18 = 0. P p. L and meet each other at the point ( 0, ) The lines L 1 and L 2 cross the x axis at the points Q and R, respectively. a) Find the value of p. b) Determine an equation for L 2. c) Show that the area of the triangle PQR is square units. C1F, p = 9, 3x + 2y = 27

34 Question 36 (***) The straight line 1 l passes through the points A ( 1,2 ) and ( 7, 2) B. a) Determine an equation for l 1, giving the answer in the form ax + by = c, where a, b and c are integers. The straight line l 2 passes through B and is perpendicular to l 1. b) Find an equation for l 2. l 2 meets the straight line with equation x = 1 at the point C. c) Calculate the area of the triangle ABC. C1K, 2x + 3y = 8, 3x 2y 25 = 0, area = 39

35 Question 37 (***) The straight line 1 l passes through the points A( 8, 2) and ( 10,1) B. a) Determine an equation of l 1 giving the answer in the form ax + by + c = 0, where a, b and c are integers. The straight line 2 lines meet at the point D. l has gradient 8 and passes through the point C ( 2,2) b) Show that D lies on the y axis. The point E has coordinates ( 1, 6).. The two c) Show clearly that EA = ED. 3x 2y 28 = 0

36 Question 38 (***) The straight lines with equations intersect at the point P( 2, ) y = 3x + c and y = 2x + 7 k, where c and k are constants. Find the value of c and the value of k. C1P, c = 5, k = 11 Question 39 (***) The straight line AB has equation 3x 4y 9 6,4. + = and C is the point ( ) The straight line BC is perpendicular to AB. Find, showing a clear method, a) an equation for the straight line BC, giving the answer in the form ax + by = c, where a, b and c are integers. b) the coordinates of B. 4x 3y 12 =, B ( 3,0)

37 Question 40 (***) y l 1 l 2 A B O D x C The straight line l 1 has equation 3x 2y + 6 = 0, and crosses the y axis at the point B. a) Find the gradient of l 1. The straight line l 2 intersects 1 point D. l at the point ( 2,6) b) Given that BAD = 90, find an equation of l 2. A and crosses the x axis at the The point C is such so that ABCD is a rectangle, as shown in the figure above. c) Calculate the area of the rectangle ABCD. C1V, gradient = 3, 3y + 2x = 22, area = 39 2

38 Question 41 (***) The straight line 1 l passes through the points A ( 3,20) and ( 13,0) B. The straight line l 2 has gradient 1 3 and passes through the point C ( 0,5 ). The point D is the intersection of l 1 and l 2. Show that the length of AD is k 5, where k is an integer. MP1-A, k = 6

39 Question 42 (***+) The points A and B have coordinates ( 4,4) and ( 2,6 ), respectively. The straight line L 1 passes through the point B and is perpendicular to the straight line which passes through A and B. a) Find an equation of L 1. L 1 meets the y axis at the point C. b) Show by calculation that AB = BC. The quadrilateral ABCD is a square whose diagonals intersect at the point M. c) Determine i. the coordinates of M. ii. the coordinates of D. C1R, y 12 3x =, M ( 2,8), D( 6,10)

40 Question 43 (***+) The straight line 1 l crosses the coordinate axes at the points A( 6,0) and ( 0,18) B. a) Determine an equation for l 1, giving the answer in the form y = mx + c, where m and c are constants. The straight line l 2 has equation x 7y = 14, and meets l 1 at the point D. b) Find the coordinates of D. l 2 meets the y axis at the point C. c) Show clearly that CAD = 90. d) Calculate the area of the triangle CAD. y = 3x + 18, D( 7, 3), area = 10

41 Question 44 (***+) The straight line L passes through the points A( 2,2) and ( 1,3) B. a) Find an equation for L. The point C has coordinates ( 3,k ). Given that the angle ABC is 90, find b) the value of k. c) the area of the triangle ABC. x 3y + 8 = 0, k = 3, area = 10

42 Question 45 (***+) The straight line segment joining the points with coordinates ( 7,k ) and ( 2, 11) where k is a constant, has gradient 2. a) Determine the value of k., The midpoint of the straight line segment joining the points with coordinates ( a, 3) and ( 4,1 ) has coordinates ( 9,b ), where a and b are constants. b) Find the value of a and the value of b. The straight line segment joining the points with coordinates ( 7,7) and ( 3,c) where c is a constant, has length 17. c) Determine the possible values of c., C1W, k = 1, a = 14, b = 1, c = 6, 8

43 Question 46 (***+) The points A( 1,3), B ( 3,1) and ( 5,5) C are given. a) Show that AB is perpendicular to BC. b) Find an equation for the straight line which passes through B and C. The straight line through A and C has equation x 3y + 10 = 0. The midpoint of AB is the point M. c) Determine an equation of the straight line which passes through M and is parallel to the straight line through A and C. The straight line which passes through M and is parallel to the straight line through A and C, meets BC at the point P. ( 4) d) State the coordinates of P. ( 2) C1A, y = 2x 5, 3y x 5 = +, P ( 4,3)

44 Question 47 (***+) 6,k, The points A, B, C and D have coordinates ( 0,4 ), ( 2,8 ), ( 3,0 ) and ( ) respectively, where k is a constant. The straight line L 1 passes through the points A and B, and the straight line L 2 passes through the points C and D. a) Given that L 1 is parallel to L 2, find an equation for L 2. b) Find the value of k. The straight line L 3 passes through the point A is perpendicular to L 2. c) Determine an equation for L 3. d) Show that L 2 and L 3 meet at the point with coordinates ( 4,2 ). y = 2x 6, k = 6, x + 2y = 8

45 Question 48 (***+) The straight line 1 l passes through the points A ( 0,3) and ( 12,9) B. a) Find an equation for l 1. The straight line 2 l passes through the point ( 11,1) C and is perpendicular to l 1. The two lines intersect at the point P. b) Calculate the coordinates of P. c) Determine the length of CP. d) Hence, or otherwise, show that the area of the triangle APC is 30 square units. C1L, x 2y =, ( 8,7) P, CP = 3 5

46 Question 49 (***+) The straight lines l 1 and l 2 have respective equations x + 4y + 5 = 0 and y = 2x 2. These two lines intersect at the point P. a) Sketch l 1 and l 2 on the same diagram, showing clearly all the points where each of these lines meet the coordinate axes. b) Calculate the exact coordinates of P. c) Determine an equation of the straight line which passes through P, and is perpendicular to l 1. Give the answer in the form ax + by = c, where a, b and c are integers. C1O, 1 (, 4 ) P, 12x 3y = 8 3 3

47 Question 50 (***+) The points A, B and C have coordinates ( 6,5), ( 0,7 ) and ( 8,3 ), respectively. The straight line L 1 is parallel to BC and passes through the point A. a) Show that an equation for L 1 is x + 2y = 4. The straight line L 2 passes through the point C is perpendicular to BC. b) Find an equation for L 2. The lines L 1 and L 2 meet at the point D. c) Show that the distance BD is 10 units. y = 2x 13

48 Question 51 (***+) The points A( 1, 1), B ( 8,2) and ( 0,1) C are given. a) Find an equation for the straight line L 1, which passes through A and B. The straight line L 2 has gradient 3 and passes through C. L 1 and L 2 intersect at the point D. b) Find the exact coordinates of D. c) Find the exact length of CD, giving the answer in the form of k 10, where k is a constant. d) Hence, or otherwise, show clearly that the area of the triangle ABC is 7.5 square units. =, 1 (, 1 ) 3y x 2 D, CD =

49 Question 52 (***+) y A M O B C x The figure above shows the points A ( 0,12), B, C ( 10,2) and ( 2,6) M. a) Given that M is the midpoint of AB, state the coordinates of B. b) Find the exact length of BC. c) Given that ABC = 90, find the area of the triangle ABC, giving the final answer as an integer. The straight line through M and perpendicular to AB meets AC at the point N. d) Determine the area of the trapezium MBCN. C1N, B ( 4,0), 40 BC =, area of ABC = 40, area of MBCN = 30

50 Question 53 (***+) The straight line L 1 has gradient 3 and y intercept of 8. The straight line L 2 is perpendicular to L 1 and passes through the point ( 7,3 ). The point P is the intersection of L 1 and L 2. a) Write down an equation for L 1. b) Find the coordinates of P. L 1 and L 2 meet the x axis at the points A and B, respectively. c) Determine the exact area of the triangle APB. y = 3x 8, P ( 4,4), area = 80 3

51 Question 54 (***+) The straight line 1 L passes through the points A ( 1,4 ) and ( 3,9) B. a) Find an equation for L 1, giving the answer in the form ax + by + c = 0, where a, b and c are integers. The straight line L 2 is perpendicular to L 1 and passes through the points B and C. Given the point C has coordinates ( 13,k ), find b) the value of k. c) the area of the triangle ABC. 5x 2y + 3 = 0, k = 5, area = 29

52 Question 55 (***+) The straight line PQ has equation 5x + 3y = 18 and P is the point with coordinates ( 9, 9). a) Find an equation for the straight line that is perpendicular to the line PQ and passing through the point P, giving the answer in the form ax + by = c, where a, b and c are integers. The straight line QR has equation y = x 6. b) Find the exact coordinates of Q. The point M ( 7, 3) is the midpoint of PT. c) Find the coordinates of T. =, 9 (, 3 ) 3x 5y 72 Q, T ( 5,3) 2 2

53 Question 56 (***+) The straight line 1 l passes through the points A( 2, 3) and ( 1, 12) B. a) Find an equation for l 1. The point M is the midpoint of AB. The straight line l 2 passes through M and has gradient 3 2. b) Find an equation for l 2. l 1 and l 2 cross the y axis at the points P and Q, respectively. c) Show that the area of the triangle PMQ is y = 3x 9, 6x 4y 27 = 0

54 Question 57 (***+) The straight line l passes through the point A( a,3), where a is a constant, and is perpendicular to the line with equation 3x + 4y = 12. Given that l crosses the y axis at ( 0, 5), find the value of a. a = 6 Question 58 (***+) The points A( 3, 1), B( 4,3), C ( 3,6) and ( 4,2) quadrilateral ABCD. D form the vertices of the By calculating relevant gradients and lengths show that ABCD is a parallelogram but not a rhombus or a rectangle. proof

55 Question 59 (***+) A triangle has vertices at A ( 1,0 ), B ( 9,4) and (,6) C k. a) Given that ABC = 90, show that k = 8. b) Find an equation of the straight line BC, giving the answer in the form ax + by = c, where a, b and c are integers. The line BC meets the x axis at the point D. c) Find the area of the triangle ABD. d) Hence, or otherwise, determine the area of the triangle ABC. 2x + y = 22, area ABD = 20, area ABC = 10

56 Question 60 (***+) y D A B C x The figure above shows a trapezium ABCD. The side AB is parallel to CD and the angles BAD and ADC are both right angles. The coordinates of D are ( 4,7 ), and the straight line through A and B has equation 5x + 4y = 7. a) Show that an equation for the straight line through C and D is 5x + 4y = 48. b) Find an equation for the straight line through A and D. The straight line through B and C has equation x + 9y + 15 = 0. c) Show that the coordinates of C are ( 12, 3). C1C, 4x 5y + 19 = 0

57 Question 61 (***+) A( 2,3) B D C ( 3,0) The figure above shows a rhombus ABCD, where the vertices A and C have 3,0, respectively. coordinates ( 2,3 ) and ( ) a) Show that an equation of the diagonal BD is x 3y + 2 = 0. b) Given that an equation of the line through A and D is 3x 4y + 6 = 0, find the coordinates of D. c) State the coordinates of B. D( 2,0), B ( 7,3)

58 Question 62 (***+) The straight line 1 l passes through the point ( 10, 3) and has gradient 1 3. a) Find an equation for l 1, in the form ax + by + c = 0, where a, b and c are integers. The straight line l 2 has gradient of 2 and y intercept of 3. l 1 and l 2 intersect at the point P. b) Determine the coordinates of P. l 1 meets the y axis at the point Q. c) Calculate the exact area of the triangle OPQ, where O is the origin. x 3y 19 = 0, P( 4, 5), area = 38 3

59 Question 63 (***+) The points A and B have coordinates ( 4,5) and ( 0,4 ), respectively. a) Find an equation of the straight line which passes through A and B. The point C lies on the straight line through A and B, so that the distance of AB is the same as the distance of BC. b) Find the coordinates of C. x + 4y = 16, C ( 4,3)

60 Question 64 (***+) The distance of the point A from the origin O is exactly a) Given the coordinates of A are ( 3t 1, t 7) the possible values of t. It is further given that A lies on the straight line with equation 5y + 2x = 18. b) Calculate the distance AB if B has coordinates ( 6,4 )., where t is a constant, determine k = 3 or 5, AB = 10

61 Question 65 (***+) y B L 1 A E L 2 O C D x L 4 L 3 The figure above shows the points A ( 0,5), B ( 4,7), C ( 1,0 ) and ( 3,1) D. The straight line L 1 passes through A and B, and the straight line L 2 passes through C and D. a) Find an equation for L 1. b) Show that L 2 is parallel to L 1. The straight line L 3 passes through A and D, and the straight line L 4 passes through B and C. The lines L 3 and L 4 intersect at the point E. c) Determine the ratio of the area of the triangle ABE to that of ECD. The individual calculations of these areas are not needed for this part. C1P, y = 1 x + 5, 4 :1 2

62 Question 66 (***+) The straight line l 1 with equation 3x 2y = 5 crosses the y axis at the point P, and the point Q has coordinates ( 6, 2). a) Find the exact coordinates of the midpoint of PQ. The straight line l 2 passes through the point Q and is perpendicular to l 1. The two lines intersect at the point R. b) Find, showing a clear method, i. an equation for l 2, giving the answer in the form ax + by = c, where a, b and c are integers. ii.... the exact coordinates of R. ( 3, 9 ) M, 2x 3y =, R 27 8 (, ) 13 13

63 Question 67 (***+) y C D B O P A x The figure above shows the square ABCD, where the vertices of A and D lie on the x axis and the y axis, respectively. The point P lies on the y axis so that PAB is a straight line. Given that the equation of the straight line through A and D is y + 2x = 6, show clearly that the distance PD is 7.5 units. C1I, proof

64 Question 68 (***+) The straight line 1 l passes through the points A( 4, 7) and ( 4,9) B. The straight line l 2 has equation and meets the l 1 at the point C. y = 1 x + 1, 2 a) Determine an equation for l 1. b) Find the coordinates of C. The straight line l 3 is the bisector of the angle formed by l 1 and l 2. c) Given that l 3 has positive gradient, determine an equation for l 3. MP1-C, y 2x 1 = +, ( 2,5) C, y = x + 3

65 Question 69 (****) A straight line L and a curve C have the following equations. 1 1 L : y = = 1 x y and C : xy + 2 = 0. a) Find the coordinates of the points of intersection between L and C. b) Sketch the graphs of L and C in the same set of axes. MP1-D, ( 2,1 ), ( 1, 2)

66 Question 70 (****) The straight line 1 point B ( 0,2). l passes through the point ( 7, 4) A and meets the y axis at the a) Determine, with full justification, whether the point P( 5, 2) lies above l 1 or below l 1. The straight line 2 l passes through the point ( 1, 10) C and meets the l 1 at B. b) Determine, with full justification, whether the point 1 (, 5 ) outside the triangle ABC. Q lies inside or 2 MP1-B, P is above l 1, Q is outside ABC

67 Question 71 (****) The points A, B and C have coordinates ( 2,0 ), ( 7,2) and ( ) 5,5, respectively. The straight line through A and perpendicular to BC, intersects BC at the point D. Find the coordinates of D. D ( 1,4 ) Question 72 (****) The points A ( 3,7), B ( 5,2) and ( 11,4 ) C are given. Calculate the area of the triangle ABC. C1Y, area = 17

68 Question 73 (****) The points A and B have coordinates ( 1,4 3 ) and ( 3+ 3,3), respectively. a) Find an equation for the straight line L which passes through A and B, y = 3 x + k, where k is an integer. giving the answer in the form ( ) L meets the x axis at the point C. b) Determine the length of AC. c) Calculate the acute angle between L and the x axis. SYN-D, y = 3x + 3 3, AC = 8, 60

69 Question 74 (****) The points A and C are the diagonally opposite vertices of a square ABCD. The straight line l 1 with equation 3x 2y = 24, meets the x and y axes at A and C, respectively. The straight line l 2 passes through B and D. Determine an equation of l 2. SYN-E, 2x + 3y + 10 = 0

70 Question 75 (****) y B l 2 Q P l 1 O A x The straight line 1 l passes though the points A ( 15,0) and ( 0,30) B. a) Determine an equation for l 1. The straight line l 2 is perpendicular to 1 k is a positive constant. The point P is the intersection between l 1 and l 2. l and passes though the point Q( 0, ) k, where b) Find, in terms of k, the x coordinate of P. c) Given further that the area of the triangle OQP is 25, where O is the origin, determine the possible area of the quadrilateral AQPA. MP1-M, y = 30 2x, x = 12 2 k, area = 40 or 100 5

71 Question 76 (****) The straight line l passes through the point A ( 3,4) and has gradient 1 2. a) Find an equation of l, giving the answer in the form y = mx + c, where m and c are constants. b) Show that B( 3,1) also lies on l. c) Calculate, in exact surd form, the distance of AB. The point P lies on l and has x coordinate p, where p is a constant. d) Given that the distance AP is 125, determine the possible values of p. C1D, y = 1 x + 5, AB = 3 5 = 45, k = 7,13 2 2

72 Question 77 (****) The straight line l passes through the points A ( 6,0) and ( 10,8) B. a) Find an equation for l, giving the answer in the form y = mx + c, where m and c are constants. The midpoint of OA is M and the midpoint of OB is N, where O is the origin. b) State the coordinates of M and N. c) Determine the area of the trapezium ABNM. C1Z, y 2x 12 =, M ( 3,0), ( 5,4) N, area = 18

73 Question 78 (****) The straight line l 1 has equation 2x + y 18 = 0 and crosses the x axis at the point P. The straight line l 2 is parallel to 1 Q. l and passes through the point ( 4,6) The point R is the x intercept of l 2. a) Determine the coordinates of P and R. The point S lies on l 1 so that RS is perpendicular to l 1. b) Show that S has coordinates ( 7,4 ). c) Calculate the area of the triangle PRS. P ( 9,0), ( 1,0 ) R, area = 20

74 Question 79 (****) y C B D not drawn to scale O A x The figure above shows the rectangle ABCD, with A ( 5,0), B ( 0,12) and ( 24, ) C k. a) Find the gradient of the straight line BC. b) Show clearly that k = 22. c) Calculate the area of the rectangle ABCD. d) Determine the coordinates of D. 5 12, area 338 =, D ( 29,10)

75 Question 80 (****) y B l 2 C l 1 D NOT DRAWN TO SCALE O A x The points A ( 1,2 ), B ( 3,8) and ( 13,8) C are shown in the figure above. The straight lines l 1 and l 2 are the perpendicular bisectors of the straight line segments AB and BC, respectively. a) Find an equation for l 1. The point D is the intersection of l 1 and l 2. b) Show clearly that i. the coordinates of D are ( 8,3 ). ii. the point D is equidistant from A, B and C. x + 3y = 17

76 Question 81 (****) P y O Q R 2 y = x 2x + 3 x The figure above shows the curve C with equation 2 y = x 2x + 3. The points P( 1,6 ), ( 0,3) Q and R all lie on C. Given that PQR = 90, determine the exact coordinates of R. SYN-C, R 7 34 (, ) 3 9

77 Question 82 (****) The straight lines l 1 and l 2, with respective equations y = 3x and 3x + 2y = 13, intersect at the point P. a) Determine the gradient of l 2. b) Find the exact coordinates of P. The lines l 1 and l 2 intersect the line with equation y = 1 at the points A and B, respectively. c) Show that the area of the triangle ABP is square units. 3 2 m =, P (, ) 9 3

78 Question 83 (****) The points A, B and C have coordinates ( 1,5 ), ( 2, y) and ( 2, 3), respectively. a) Find, in terms of y, the gradient of BC. The angle ABC is 90. b) Determine the possible values of y. C1X, y + 3, y = 1, y = 3 4

79 Question 84 (****) The rectangle ABCD has three of its vertices located at A ( 5,10), ( 3, ) C ( 9,2), where k is a constant. a) Show that ( k )( k ) = 0. B k and and hence determine the two possible values of k. It is further given that the rectangle ABCD reduces to a square, for one of the two values of k found in part (a). For the square ABCD... b)... determine its area. c)... state the coordinates of D. SYN-B, k = 8, area 40 =, D ( 11,8 )

80 Question 85 (****+) The points A and B have coordinates ( 8,2 ) and ( 11,3 ), respectively. The point C lies on the straight line with equation y + x = 14. Given further that the distance AC is twice as large as the distance AB, determine the two possible sets of coordinates of C. SYN-F, C ( 6,8) C ( 14,0)

81 Question 86 (****+) The points A and B have coordinates ( 3,10) and ( 9,6 ), respectively. a) Find an equation for the straight line L 1 which passes through the point B and is perpendicular to AB, in the form y = mx + c, where m and c are constants. L 1 crosses the x axis at the point C. b) Determine the value of tanθ, where θ is the angle that BC makes with the positive x axis. c) Find an equation for the straight line L 2 which passes through the point C and is parallel to AB, in the form ax + by = c, where a, b and c are integers. The point D is such so that ABCD is a rectangle. d) Show that the coordinates of D are ( 5,4). e) Find the area of the rectangle ABCD. y = 3x 21, tanθ = 3 x + 3y = 7 area = 80

82 Question 87 (****+) The points A and C have coordinates ( 3,2 ) and ( 5,6 ), respectively. a) Find an equation for the perpendicular bisector of AC, giving the answer in the form ax by c + =, where a, b and c are integers. The perpendicular bisector of AC crosses the y axis at the point B. The point D is such so that ABCD is a rhombus. b) Show that the coordinates of D are ( 8,2 ). c) Calculate the area of the rhombus ABCD. x + 2y = 12, area = 20

83 Question 88 (****+) The straight line l passes through the point P ( 4,5) and has gradient 3. The point Q also lies on l so that the distance PQ is Determine the coordinates of the two possible positions of Q. Q( 7,14) or Q( 1, 4)

84 Question 89 (****+) A y B l 1 O D C x l 2 The figure above shows a parallelogram ABCD. The straight line 1 l passes through A( 1,3) and ( 4,4) B. a) Find an equation for l 1. Give the answer in the form ax + by + c = 0, where a, b and c are integers. The points C and D lie on the straight line l 2, which has equation 5y x + 10 = 0. b) Show that the distance between l 1 and l 2 is k, where k is an integer. c) Hence, find the area of the parallelogram ABCD. C1E, 5y x 16 = 0, k = 26, area = 26

85 Question 90 (****+) y B A D C O x The figure above shows a trapezium OABC, where O is the origin, whose side AB is parallel to the side OC. The diagonal AC is horizontal and the points A and B have coordinates ( 0,6 ) and ( 8,8 ), respectively. The diagonals of the trapezium meet at the point D. Show by direct area calculations that the area of the triangle BCD is equal to the area of the triangle OAD. C1H, proof

86 Question 91 (****+) y C not drawn to scale D B A O E x A parallelogram has vertices at A( 3, 2), B( 4,2), C ( 3,8) and ( 4,4) D. a) Show that ABD = 90 and hence find the area of the parallelogram ABCD. The side CD is extended so that it meets the x axis at the point E. b) Find the coordinates of E. c) Show that EB and AD bisect each other. d) By considering two suitable congruent triangles and without any direct area calculations, show that the area of the triangle EBC is equal to the area of the parallelogram ABCD. area 34 =, E ( 5,0)

87 Question 92 (****+) The straight line 1 constant. l passes through the points A ( 2,1) and B( k,8), where k is a a) Given the gradient of l 1 is 7 2 show that k = 4. b) Find an equation for l 1. The straight line l 2 is parallel to the y axis and passing through A. The point C lies on l 2 so that the length of BC is exactly 2 2. c) Find the possible coordinates of C. d) Determine the largest possible area of the triangle ABC. C1T, 2y 7x 12 =, ( 2,6) or ( 2,10) C C, area = 9

88 Question 93 (****+) The straight lines L 1 and L 2 have respective equations 4x + 2y = a and 5x + 4y = b. It is given that L 1 and L 2 meet at the point P. Express a in terms of b, given further that P lies in the second quadrant and is equidistant from the coordinate axes. SP-Z, a = 2b

89 Question 94 (****+) The points A and B have coordinates ( 0, 4) and ( 3, 2), respectively. a) Determine an equation for the straight line l which passes through the points A and B, giving the answer in the form ax + by + c = 0, where a, b and c are integers. The point C lies on l, so that the distance AC is 3 13 units. b) Show, by a complete algebraic solution, that one possible set of coordinates for C are (9,2) and find the other set. C1U, 2x 3y 12 0 =, C ( 9, 10)

90 Question 95 (****+) The point A lies on the straight line L with equation y = 2x + 3. The point B has coordinates ( 4,1 ) and the point C is the reflection of A about L. Determine the possible coordinates of A, given that AB is perpendicular to AC. SP-N, A( 2,7 ) or A( 2, 1)

91 Question 96 (****+) The points A( 3,9), B ( 4,5) and ( 7, 6) C are three vertices of the kite ABCD. Determine the coordinates of D. SP-D, D( 2,1)

92 Question 97 (****+) The point A ( 1,3) is one of the vertices of a rhombus whose centre is ( 3,7) B. One of the sides of the rhombus lies on the straight line with equation y = 5 2x. Determine the coordinates of the other three vertices of this rhombus. SP-G, ( 5,11 ), ( 7, ), (, )

93 Question 98 (****+) The point A has coordinates ( 2,1). a) Find the coordinates of the point of reflection of A about the straight line with equation 3x + y = 12. The point P, whose coordinates are ( 4,2 ), is rotated about A by 90 anticlockwise, onto the point Q. b) Determine the coordinates of Q. MP1-U, ( 8.2,4.4 ), Q( 3,7)

94 Question 99 (*****) Find the shortest distance between the parallel lines with equations x + 2y = 10 and x + 2y = 20. [You may not use a standard formula which determines the shortest distance of a point from a straight line in this question] C1S, 2 5

95 Question 100 (*****) The straight line L 1, has gradient m, 0 Another straight line L 2 is perpendicular to 1 k 3. The point C is the intersection of L 1 and L 2. A. m > and passes through the point ( 2,3) B k, L and passes through the point ( 2, ) Determine the y coordinate of C, in terms of k and m, and given further that the triangle ABC is isosceles, prove that m = 1. SP-H, km y = m

96 Question 101 (*****) The variables x and y are such so that ax + by = c, where a, b and c are non zero constants. Show that the minimum value of x y is a c b. SP-Z, proof

97 Question 102 (*****) y B A C O x The figure above shows an isosceles trapezium OABC, where O is the origin. It is further given that the coordinates of A are ( 0,6 ), the sides AB and OC are parallel, OA = BC, the diagonal AC is parallel to the x axis. Determine, in exact simplified surd form, the coordinates of B. MP1-S, B ( , )

98 Question 103 (*****) The straight line 1 l passes though the points A ( 30,0) and ( 0,10) B. The straight line l 2 is perpendicular to l 1. The point P is the intersection between l 1 l 2, and the point Q is the point where l 2 meets the y axis. Given further that the area of the triangle OQP is 2.4, where O is the origin, find the possible area of the quadrilateral AQPA. SPX-A, area = or 149.4

99 Question 104 (*****) The straight line L 1 has equation The straight line 2 y = 6 2x, L passes through the point ( 2,7) A and meets L 1 at the point B. Given that L 1 and L 2 intersect each other at 45, determine the coordinates of B. SP-F, B ( 1,4 )

100 Question 105 (*****) The straight lines L 1 and L 2 have respective equations The point A has coordinates ( 4,2 ). y = 3x 10 and y = 5x 4. The point B lies on L 1, so that the midpoint M of the straight line segment AB, lies on L 2. Determine the coordinates of B and the coordinates of M. SP-A, B 5 (, ), M (, )

101 Question 106 (*****) The straight line l has equation ( ) ( ) 2 + a x + 2 a y = 2 5a, where a is a constant. Show that l passes through a fixed point P for all values of a. SP-C, proof

102 Question 107 (*****) A family of straight lines, has equation ( ) y = m x 1 + 2, x R, where m is a parameter. From the above family of straight lines, determine the equations of any straight lines whose distance from the origin O is 1 unit. = , x = 1 4 SP-K, y ( x )

103 Question 108 (*****) The straight lines l 1 and l 2 have respective equations The straight line 3 y = x and x + y = 2. P h k. l is parallel to the x axis and passes through the point (, ) It is further given that the point B is the intersection of l 1 and l 2, the point A is the intersection of l 1 and l 3 and the point C is the intersection of l 2 and l 3, Find the locus of P if the area of the triangle ABC is 2 9h. SP-U, y = 1± 3x

104 Question 109 (*****) The straight parallel lines l 1 and l 2 have respective equations The straight line 3 points A and B respectively. y = 3x + 2 and y = 3x 4. l, passing through the point ( 9,13) P, intersects l 1 and l 2 at the Given that AB = 18 determine the possible equation of l 3. SP-R, y = x + 4, y = 76 7x

105 Question 110 (*****) The straight parallel lines l 1 and l 2 have respective equations y = 3x + 5 and y = 3x + 2. Two more straight lines, both passing through the origin O, intersect l 1 and l 2 forming a trapezium ABCD. The trapezium ABCD is isosceles with AB = CD = 3 5. Determine the area of this trapezium. SP-V, area = 14.7

106 Question 111 (*****) Show that the area of the triangular region bounded by 2 2 x y + x + 3y = 2 and 2x y = 2, is square units. SP-L, proof

107 Question 112 (*****) A family of straight lines has equation ( ) ( ) a + a + 3 x + a a 3 y = 7a 3a + 1, where a is a parameter. The point Q has coordinates ( 20,7 ). Show that this family of lines passes through a fixed point P for all values of a, and hence determine the equation of a straight line from this family of straight lines, which is furthest away from Q. SP-S, y = 23 9x

108 Question 113 (*****) Two parallel straight lines, L 1 and L 2, have respective equations y = 2x + 5 and y = 2x 1. L 1 and L 2, are tangents to a circle centred at the point C. A third line L 3 is perpendicular to L 1 and L 2, and meets the circle in two distinct points, A and B. Given that 3 coordinates of C. L passes through the point ( ) 9,0, find, in exact simplified surd form, the SP-Q, C 1 ( ), ( ) 10 5

CIRCLE COORDINATE GEOMETRY

CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle

Geometry Module 4 Unit 2 Practice Exam

Name: Class: Date: ID: A Geometry Module 4 Unit 2 Practice Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which diagram shows the most useful positioning

C1: Coordinate geometry of straight lines

B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On

San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS

San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS Recall that the bisector of an angle is the ray that divides the angle into two congruent angles. The most important results about angle bisectors

QUADRILATERALS CHAPTER 8. (A) Main Concepts and Results

CHAPTER 8 QUADRILATERALS (A) Main Concepts and Results Sides, Angles and diagonals of a quadrilateral; Different types of quadrilaterals: Trapezium, parallelogram, rectangle, rhombus and square. Sum of

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

Section 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18

Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,

0810ge. Geometry Regents Exam 0810

0810ge 1 In the diagram below, ABC XYZ. 3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm. Which two statements identify

CHAPTER 8 QUADRILATERALS 8.1 Introduction You have studied many properties of a triangle in Chapters 6 and 7 and you know that on joining three non-collinear points in pairs, the figure so obtained is

Equation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1

Chapter H2 Equation of a Line The Gradient of a Line The gradient of a line is simpl a measure of how steep the line is. It is defined as follows :- gradient = vertical horizontal horizontal A B vertical

Analytical Geometry (4)

Analytical Geometry (4) Learning Outcomes and Assessment Standards Learning Outcome 3: Space, shape and measurement Assessment Standard As 3(c) and AS 3(a) The gradient and inclination of a straight line

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

116 Chapter 6 Transformations and the Coordinate Plane

116 Chapter 6 Transformations and the Coordinate Plane Chapter 6-1 The Coordinates of a Point in a Plane Section Quiz [20 points] PART I Answer all questions in this part. Each correct answer will receive

www.sakshieducation.com

LENGTH OF THE PERPENDICULAR FROM A POINT TO A STRAIGHT LINE AND DISTANCE BETWEEN TWO PAPALLEL LINES THEOREM The perpendicular distance from a point P(x 1, y 1 ) to the line ax + by + c 0 is ax1+ by1+ c

DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent

SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY. 2.1 Use the slopes, distances, line equations to verify your guesses

CHAPTER SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY For the review sessions, I will try to post some of the solved homework since I find that at this age both taking notes and proofs are still a burgeoning

Co-ordinate Geometry THE EQUATION OF STRAIGHT LINES

Co-ordinate Geometry THE EQUATION OF STRAIGHT LINES This section refers to the properties of straight lines and curves using rules found by the use of cartesian co-ordinates. The Gradient of a Line. As

Geometry Regents Review

Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest

Straight Line. Paper 1 Section A. O xy

PSf Straight Line Paper 1 Section A Each correct answer in this section is worth two marks. 1. The line with equation = a + 4 is perpendicular to the line with equation 3 + + 1 = 0. What is the value of

1. An isosceles trapezoid does not have perpendicular diagonals, and a rectangle and a rhombus are both parallelograms.

Quadrilaterals - Answers 1. A 2. C 3. A 4. C 5. C 6. B 7. B 8. B 9. B 10. C 11. D 12. B 13. A 14. C 15. D Quadrilaterals - Explanations 1. An isosceles trapezoid does not have perpendicular diagonals,

Chapter 6 Quiz. Section 6.1 Circles and Related Segments and Angles

Chapter 6 Quiz Section 6.1 Circles and Related Segments and Angles (1.) TRUE or FALSE: The center of a circle lies in the interior of the circle. For exercises 2 4, use the figure provided. (2.) In O,

Algebra III. Lesson 33. Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids

Algebra III Lesson 33 Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids Quadrilaterals What is a quadrilateral? Quad means? 4 Lateral means?

206 MATHEMATICS CIRCLES

206 MATHEMATICS 10.1 Introduction 10 You have studied in Class IX that a circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre). You have

A. 3y = -2x + 1. y = x + 3. y = x - 3. D. 2y = 3x + 3

Name: Geometry Regents Prep Spring 2010 Assignment 1. Which is an equation of the line that passes through the point (1, 4) and has a slope of 3? A. y = 3x + 4 B. y = x + 4 C. y = 3x - 1 D. y = 3x + 1

Circle Name: Radius: Diameter: Chord: Secant:

12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane

http://www.castlelearning.com/review/teacher/assignmentprinting.aspx 5. 2 6. 2 1. 10 3. 70 2. 55 4. 180 7. 2 8. 4

of 9 1/28/2013 8:32 PM Teacher: Mr. Sime Name: 2 What is the slope of the graph of the equation y = 2x? 5. 2 If the ratio of the measures of corresponding sides of two similar triangles is 4:9, then the

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, June 17, 2010 1:15 to 4:15 p.m., only Student Name: School Name: Print your name and the name of your

Quadrilateral Geometry. Varignon s Theorem I. Proof 10/21/2011 S C. MA 341 Topics in Geometry Lecture 19

Quadrilateral Geometry MA 341 Topics in Geometry Lecture 19 Varignon s Theorem I The quadrilateral formed by joining the midpoints of consecutive sides of any quadrilateral is a parallelogram. PQRS is

6-5 Rhombi and Squares. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.

ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. 1. If, find. A rhombus is a parallelogram with all four sides congruent. So, Then, is an isosceles triangle. Therefore, If a parallelogram

Angles in a Circle and Cyclic Quadrilateral

130 Mathematics 19 Angles in a Circle and Cyclic Quadrilateral 19.1 INTRODUCTION You must have measured the angles between two straight lines, let us now study the angles made by arcs and chords in a circle

Geometry in a Nutshell

Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2009 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2009 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

GEOMETRIC MENSURATION

GEOMETRI MENSURTION Question 1 (**) 8 cm 6 cm θ 6 cm O The figure above shows a circular sector O, subtending an angle of θ radians at its centre O. The radius of the sector is 6 cm and the length of the

Tips for doing well on the final exam

Name Date Block The final exam for Geometry will take place on May 31 and June 1. The following study guide will help you prepare for the exam. Everything we have covered is fair game. As a reminder, topics

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, August 18, 2010 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 20, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name

Conjectures. Chapter 2. Chapter 3

Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

75 Chapter 11 11.1 Quadrilateral A plane figure bounded by four sides is known as a quadrilateral. The straight line joining the opposite corners is called its diagonal. The diagonal divides the quadrilateral

Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

39 Symmetry of Plane Figures

39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that

DIFFERENTIATION OPTIMIZATION PROBLEMS

DIFFERENTIATION OPTIMIZATION PROBLEMS Question 1 (***) 4cm 64cm figure 1 figure An open bo is to be made out of a rectangular piece of card measuring 64 cm by 4 cm. Figure 1 shows how a square of side

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

MEI STRUCTURED MATHEMATICS INTRODUCTION TO ADVANCED MATHEMATICS, C1. Practice Paper C1-C

MEI Mathematics in Education and Industry MEI STRUCTURED MATHEMATICS INTRODUCTION TO ADVANCED MATHEMATICS, C Practice Paper C-C Additional materials: Answer booklet/paper Graph paper MEI Examination formulae

SMT 2014 Geometry Test Solutions February 15, 2014

SMT 014 Geometry Test Solutions February 15, 014 1. The coordinates of three vertices of a parallelogram are A(1, 1), B(, 4), and C( 5, 1). Compute the area of the parallelogram. Answer: 18 Solution: Note

10-4 Inscribed Angles. Find each measure. 1.

Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semi-circle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what

1. Find the length of BC in the following triangles. It will help to first find the length of the segment marked X.

1 Find the length of BC in the following triangles It will help to first find the length of the segment marked X a: b: Given: the diagonals of parallelogram ABCD meet at point O The altitude OE divides

Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

Unit 3: Triangle Bisectors and Quadrilaterals

Unit 3: Triangle Bisectors and Quadrilaterals Unit Objectives Identify triangle bisectors Compare measurements of a triangle Utilize the triangle inequality theorem Classify Polygons Apply the properties

ABC is the triangle with vertices at points A, B and C

Euclidean Geometry Review This is a brief review of Plane Euclidean Geometry - symbols, definitions, and theorems. Part I: The following are symbols commonly used in geometry: AB is the segment from the

Practical Geometry CHAPTER. 4.1 Introduction DO THIS

PRACTICAL GEOMETRY 57 Practical Geometry CHAPTER 4 4.1 Introduction You have learnt how to draw triangles in Class VII. We require three measurements (of sides and angles) to draw a unique triangle. Since

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, June 19, :15 a.m. to 12:15 p.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 19, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

Triangles. (SAS), or all three sides (SSS), the following area formulas are useful.

Triangles Some of the following information is well known, but other bits are less known but useful, either in and of themselves (as theorems or formulas you might want to remember) or for the useful techniques

Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

/27 Intro to Geometry Review

/27 Intro to Geometry Review 1. An acute has a measure of. 2. A right has a measure of. 3. An obtuse has a measure of. 13. Two supplementary angles are in ratio 11:7. Find the measure of each. 14. In the

5.1 Midsegment Theorem and Coordinate Proof

5.1 Midsegment Theorem and Coordinate Proof Obj.: Use properties of midsegments and write coordinate proofs. Key Vocabulary Midsegment of a triangle - A midsegment of a triangle is a segment that connects

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, January 26, 2016 1:15 to 4:15 p.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, January 26, 2016 1:15 to 4:15 p.m., only Student Name: School Name: The possession or use of any communications

http://jsuniltutorial.weebly.com/ Page 1

Parallelogram solved Worksheet/ Questions Paper 1.Q. Name each of the following parallelograms. (i) The diagonals are equal and the adjacent sides are unequal. (ii) The diagonals are equal and the adjacent

WEDNESDAY, 2 MAY 1.30 PM 2.25 PM. 3 Full credit will be given only where the solution contains appropriate working.

C 500/1/01 NATIONAL QUALIFICATIONS 01 WEDNESDAY, MAY 1.0 PM.5 PM MATHEMATICS STANDARD GRADE Credit Level Paper 1 (Non-calculator) 1 You may NOT use a calculator. Answer as many questions as you can. Full

Name: Class: Date: Quadrilaterals Unit Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. ( points) In which polygon does the sum of the measures of

Shape, Space and Measure

Name: Shape, Space and Measure Prep for Paper 2 Including Pythagoras Trigonometry: SOHCAHTOA Sine Rule Cosine Rule Area using 1-2 ab sin C Transforming Trig Graphs 3D Pythag-Trig Plans and Elevations Area

IMO Geomety Problems. (IMO 1999/1) Determine all finite sets S of at least three points in the plane which satisfy the following condition:

IMO Geomety Problems (IMO 1999/1) Determine all finite sets S of at least three points in the plane which satisfy the following condition: for any two distinct points A and B in S, the perpendicular bisector

CHAPTER 7 TRIANGLES. 7.1 Introduction. 7.2 Congruence of Triangles

CHAPTER 7 TRIANGLES 7.1 Introduction You have studied about triangles and their various properties in your earlier classes. You know that a closed figure formed by three intersecting lines is called a

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

Definitions, Postulates and Theorems

Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven

CSU Fresno Problem Solving Session. Geometry, 17 March 2012

CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news

Properties of Special Parallelograms

Properties of Special Parallelograms Lab Summary: This lab consists of four activities that lead students through the construction of a parallelogram, a rectangle, a square, and a rhombus. Students then

Geometry A Solutions. Written by Ante Qu

Geometry A Solutions Written by Ante Qu 1. [3] Three circles, with radii of 1, 1, and, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to

Exercise Set 3. Similar triangles. Parallel lines

Exercise Set 3. Similar triangles Parallel lines Note: The exercises marked with are more difficult and go beyond the course/examination requirements. (1) Let ABC be a triangle with AB = AC. Let D be an

Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...

Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................

Angles that are between parallel lines, but on opposite sides of a transversal.

GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

The Triangle and its Properties

THE TRINGLE ND ITS PROPERTIES 113 The Triangle and its Properties Chapter 6 6.1 INTRODUCTION triangle, you have seen, is a simple closed curve made of three line segments. It has three vertices, three

Chapter 1: Essentials of Geometry

Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,

THREE DIMENSIONAL GEOMETRY

Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

The Distance from a Point to a Line

: Student Outcomes Students are able to derive a distance formula and apply it. Lesson Notes In this lesson, students review the distance formula, the criteria for perpendicularity, and the creation of

SHAPE, SPACE AND MEASURES

SHPE, SPCE ND MESURES Pupils should be taught to: Understand and use the language and notation associated with reflections, translations and rotations s outcomes, Year 7 pupils should, for example: Use,

Sum of the interior angles of a n-sided Polygon = (n-2) 180

5.1 Interior angles of a polygon Sides 3 4 5 6 n Number of Triangles 1 Sum of interiorangles 180 Sum of the interior angles of a n-sided Polygon = (n-2) 180 What you need to know: How to use the formula

Isosceles triangles. Key Words: Isosceles triangle, midpoint, median, angle bisectors, perpendicular bisectors

Isosceles triangles Lesson Summary: Students will investigate the properties of isosceles triangles. Angle bisectors, perpendicular bisectors, midpoints, and medians are also examined in this lesson. A

Lecture 24: Saccheri Quadrilaterals 24.1 Saccheri Quadrilaterals Definition In a protractor geometry, we call a quadrilateral ABCD a Saccheri quadrilateral, denoted S ABCD, if A and D are right angles

NCERT. Area of the circular path formed by two concentric circles of radii. Area of the sector of a circle of radius r with central angle θ =

AREA RELATED TO CIRCLES (A) Main Concepts and Results CHAPTER 11 Perimeters and areas of simple closed figures. Circumference and area of a circle. Area of a circular path (i.e., ring). Sector of a circle

CIRCUMFERENCE AND AREA OF A CIRCLE

CIRCUMFERENCE AND AREA OF A CIRCLE 1. AC and BD are two perpendicular diameters of a circle with centre O. If AC = 16 cm, calculate the area and perimeter of the shaded part. (Take = 3.14) 2. In the given

The Four Centers of a Triangle. Points of Concurrency. Concurrency of the Medians. Let's Take a Look at the Diagram... October 25, 2010.

Points of Concurrency Concurrent lines are three or more lines that intersect at the same point. The mutual point of intersection is called the point of concurrency. Example: x M w y M is the point of

Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.

1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called

POINT OF INTERSECTION OF TWO STRAIGHT LINES

POINT OF INTERSECTION OF TWO STRAIGHT LINES THEOREM The point of intersection of the two non parallel lines bc bc ca ca a x + b y + c = 0, a x + b y + c = 0 is,. ab ab ab ab Proof: The lines are not parallel

Quadrilaterals / Mathematics Unit: 11 Lesson: 01 Duration: 7 days Lesson Synopsis: In this lesson students explore properties of quadrilaterals in a variety of ways including concrete modeling, patty paper

egyptigstudentroom.com

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education *5128615949* MATHEMATICS 0580/04, 0581/04 Paper 4 (Extended) May/June 2007 Additional Materials:

2. THE x-y PLANE 7 C7

2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

Class-10 th (X) Mathematics Chapter: Tangents to Circles

Class-10 th (X) Mathematics Chapter: Tangents to Circles 1. Q. AB is line segment of length 24 cm. C is its midpoint. On AB, AC and BC semicircles are described. Find the radius of the circle which touches

Date: Period: Symmetry

Name: Date: Period: Symmetry 1) Line Symmetry: A line of symmetry not only cuts a figure in, it creates a mirror image. In order to determine if a figure has line symmetry, a figure can be divided into

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXMINTION GEOMETRY Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name

Math 531, Exam 1 Information.

Math 531, Exam 1 Information. 9/21/11, LC 310, 9:05-9:55. Exam 1 will be based on: Sections 1A - 1F. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/531fa11/531.html)

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

Chapter 6 Notes: Circles

Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment

Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid

Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid Grade level: 10 Prerequisite knowledge: Students have studied triangle congruences, perpendicular lines,

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

4.1 Euclidean Parallelism, Existence of Rectangles

Chapter 4 Euclidean Geometry Based on previous 15 axioms, The parallel postulate for Euclidean geometry is added in this chapter. 4.1 Euclidean Parallelism, Existence of Rectangles Definition 4.1 Two distinct

3. If AC = 12, CD = 9 and BE = 3, find the area of trapezoid BCDE. (Mathcounts Handbooks)

EXERCISES: Triangles 1 1. The perimeter of an equilateral triangle is units. How many units are in the length 27 of one side? (Mathcounts Competitions) 2. In the figure shown, AC = 4, CE = 5, DE = 3, and