# C1: Coordinate geometry of straight lines

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 distance between two given points find the coordinates of the mid-point of a line segment joining two given points find, use and interpret the gradient of a line segment know the relationship between the gradients for parallel and for perpendicular lines find the equations of straight lines given (a) the gradient and -intercept, (b) the gradient and a point, and (c) two points verif, given their coordinates, that points lie on a line find the coordinates of a point of intersection of two lines find the fourth verte of a parallelogram given the other three. Coordinate geometr is the use of algebraic methods to stud the geometr of straight lines and curves. In this chapter ou will consider onl straight lines. Curves will be studied later.. Cartesian coordinates In our GCSE course ou plotted points in two dimensions. This section revises that work and builds up the terminolog required. Named after René Descartes ( ), a French mathematician. The vertical line is called the -ais. positive -ais nd quadrant negative -ais st quadrant positive -ais The horizontal line is called the -ais. The plane is divided up into four quadrants b etending two perpendicular lines called the coordinate aes and. rd quadrant 4th quadrant The point, where the coordinate aes cross, is called the origin. negative -ais

2 B_Chap0_08-05.qd 5/6/04 0:4 am Page 9 C: Coordinate geometr of straight lines 9 In the diagram below, the point P lies in the first quadrant. Its distance from the -ais is a. Its distance from the -ais is b. We sa that: the -coordinate of P is a the -coordinate of P is b. We write P is the point (a, b) or simpl the point P(a, b). Q is in the second quadrant. The -coordinate is negative and the -coordinate is positive. The point P has Cartesian coordinates (a, b) Q c a P Coordinates of Q are (c, d) d b f h The origin is (0, 0) R e R is the point (e, f ) g S (g, h) The diagram below shows part of the Cartesian grid, where L is the point (, ). The -coordinate of L is and the -coordinate of L is. An point on the -ais has its -coordinate equal to 0. L (, ) M (0, ) J (, 0) K (, 0) 4 4 An point on the -ais has its -coordinate equal to 0. J and K are points on the -ais. N (0, )

3 B_Chap0_08-05.qd 5/6/04 0:4 am Page 0 0 C: Coordinate geometr of straight lines Worked eample. Draw coordinate aes and and plot the points A(, ), B(, ), C(, 4) and D(, 4). The line joining the points C and D crosses the -ais at the point E. Write down the coordinates of E. The line joining the points A and C crosses the -ais at the point F. Write down the coordinates of F. B(, ) 5 4 A(, ) D(, 4) 5 C(, 4) E is on the -ais so its -coordinate is 0. E(0, 4). F is on the -ais so its -coordinate is 0. F(, 0).. The distance between two points The following eample shows how ou can use Pthagoras theorem which ou studied as part of our GCSE course, in order to find the distance between two points. Worked eample. Find the distance AB where A is the point (, ) and B is the point (4, 4). First plot the points, join them with a line and make a right-angled triangle ABC. The distance AC 4. The distance BC 4. Using Pthagoras theorem AB. 9 4 AB From the diagram in Worked eample., ou can see that the distance between the points C(, 4) and D(, 4) is units and the distance between the points A(, ) and C(, 4) is 6 units, but how do ou find the distance between points which do not lie on the same horizontal line or on the same vertical line? 4 A 4 AB.6 (to two significant figures) but since a calculator is not allowed in the eamination for this unit ou should leave our answers in eact forms. Where possible surd answers should be simplified. B C

4 B_Chap0_08-05.qd 5/6/04 0:4 am Page C: Coordinate geometr of straight lines We can generalise the method of the previous eample to find a formula for the distance between the points P(, ) and Q(, ). The distance PR. The distance QR. Q(, ) Using Pthagoras theorem PQ ( ) ( ) PQ [( ) ( ) ]. P(, ) R The distance between the points (, ) and (, ) is ( ) ( ). Worked eample. The point R has coordinates (, ) and the point S has coordinates (5, 6). (a) Find the distance RS. (b) The point T has coordinates (0, 9). Show that RT has length k, where k is an integer. An integer is a whole number. (a) For points R and S: the difference between the -coordinates is 5 () 6 the difference between the -coordinates is 6 8. RS (6) (8) (b) RT () (7) Worked eample.4 The distance MN is 5, where M is the point (4, ) and N is the point (a,a). Find the two possible values of the constant a. If ou draw a diagram ou will probabl find ourself dealing with distances rather than the difference in coordinates in order to avoid negative numbers. This is not wrong. f course ou could have considered the differences as 5 6 and (6) 8 and obtained the same answer. Used a b a b. See section.4. You ma prefer to draw a diagram but it is also good to learn to work algebraicall.

5 B_Chap0_08-05.qd 5/6/04 0:4 am Page C: Coordinate geometr of straight lines The difference between the -coordinates is a 4. The difference between the -coordinates is a () a. MN (a 4) (a ) 5 (a 4) (a ) 5 a 8a 6 4a 8a 4 5 5a 0 a a. Using ( ). Notice that this equation does have two solutions. EXERCISE A Find the lengths of the line segments joining: (a) (0, 0) and (, 4), (b) (, ) and (5, ), The identit (k) (4k) (5k) ma be useful in (j), (k) and (l). (c) (0, 4) and (5, ), (d) (, ) and (, 6), (e) (4, ) and (, 0), (f) (, ) and (6, ), (g) (, 7) and (, ), (h) (, 0) and (6, ), (i) (.5, 0) and (.5, 0) (j) (.5, 4) and (, 6), (k) (8, 0) and (,.5), (l) (.5, ) and (4, 8). Calculate the lengths of the sides of the triangle ABC and hence determine whether or not the triangle is right-angled: (a) A(0, 0) B(0, 6) C(4, ), (b) A(, 0) B(, 8) C(7, 6), (c) A(, ) B(, 4) C(0, 7). From question onwards ou are advised to draw a diagram. In longer questions, the results found in one part often help in the net part. The vertices of a triangle are A(, 5), B(0, ) and C(4, ). B writing each of the lengths of the sides as a multiple of, show that the sum of the lengths of two of the sides is three times the length of the third side. 4 The distance between the two points A(6, p) and B(p, ) is 55. Find the possible values of p. 5 The vertices of a triangle are P(, ), Q(, 0) and R(4, 0). (a) Find the lengths of the sides of triangle PQR. (b) Show that angle QPR 90. (c) The line of smmetr of triangle PQR meets the -ais at point S. Write down the coordinates of S. (d) The point T is such that PQTR is a square. Find the coordinates of T.

6 B_Chap0_08-05.qd 5/6/04 0:4 am Page C: Coordinate geometr of straight lines. The coordinates of the mid-point of a line segment joining two known points From the diagram ou can see that the mid-point of the line segment joining (0, 0) to (6, 0) is (, 0) or or 0 6,0, the mid-point of the line segment joining (0, 0) to (0, 4) is (0, ) or 0,0 (4), and the mid-point of the line segment joining (0, 4) to (6, 0) is (, ) or 0 6, (4 ) 0. Going from P(, ) to Q(5, 8) ou move 4 units horizontall and then 6 verticall. If M is the mid-point of PQ then the journe is halved so to go from P(, ) to M ou move (half of 4) horizontall and then (half of 6) verticall. So M is the point (, ) or M(, 5) or M 5, 8. ( ) A(, ) M, ( ) up B(, ) mid-point 4 5 along mid-point P(, ) M mid-point of line segment is (, ) 4 along up Q(5, 8) Check that M to Q is also along and then up. 6 up ( ) along ( ) Note that ( ) ( ) which is the -coordinate of M. In general, the coordinates of the mid-point of the line segment joining (, ) and (, ) are,.

7 B_Chap0_08-05.qd 5/6/04 0:4 am Page 4 4 C: Coordinate geometr of straight lines Worked eample.5 M is the mid-point of the line segment joining A(, ) and B(, 5). (a) Find the coordinates of M. (b) M is also the mid-point of the line segment CD where C(, 4). Find the coordinates of D. (a) Using,, M is the point, 5 so M(, ). (b) Let D be the point (a, b) then the mid-point of CD is a, b 4 (, ) coordinates of M For this to be true a and b 4 a and b. The coordinates of D are (, ). 5 4 C(, 4) A(, ) B(, 5) M(, ) 4 5 D r, using the idea of journes : C to M is across and down. So M to D is also across and down giving D(, ) or D(, ). EXERCISE B Find the coordinates of the mid-point of the line segments joining: (a) (, ) and (7, ), (b) (, ) and (, ), (c) (0, ) and (6, ), (d) (, ) and (, 6), (e) (4, ) and (, 6), (f) (, ) and (6, ), (g) (, 5) and (, ), (h) (, 5) and (6, ), (i) (.5, 6) and (.5, 0) (j) (.5, ) and (4, ). M is the mid-point of the straight line segment PQ. Find the coordinates of Q for each of the cases: (a) P(, ), M(, 4), (b) P(, ), M(, ), (c) P(, ), M(, 5), (d) P(, 5), M(, 0), (e) P(, 4), M(, ), (f) P(, ), M(, 4 ). The mid-point of AB, where A(, ) and B(4, 5), is also the mid-point of CD, where C(0, ). (a) Find the coordinates of D. (b) Show that AC BD. 4 The mid-point of the line segment joining A(, ) and B(5, ) is D. The point C has coordinates (4, 4). Show that CD is perpendicular to AB.

8 B_Chap0_08-05.qd 5/6/04 0:4 am Page 5 C: Coordinate geometr of straight lines 5.4 The gradient of a straight line joining two known points The gradient of a straight line is a measure of how steep it is. () () () (a) (b) (c) These three lines slope upwards from left to right. The have gradients which are positive. Line () is steeper than line () so the gradient of () is greater than the gradient of (). These three lines are all parallel to the -ais. The are not sloping. Horizontal lines have gradient 0. These three lines slope downwards from left to right. The have gradients which are negative. change in -coordinate The gradient of the line joining two points. change in -coordinate B(, ) ( ) Change in A(, ) ( ) Change in The gradient of the line joining the two points A(, ) and B(, ) is. Lines which are equall steep are parallel. Parallel lines have equal gradients.

11 B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 8 C: Coordinate geometr of straight lines Gradient of AB 5 4. Let m be the gradient of an line perpendicular to AB then m m. The gradient of an line perpendicular to AB is. EXERCISE D Write down the gradient of lines perpendicular to a line with gradient: (a) 5, (b), (c) 4, (d), (e) 4. Two vertices of a rectangle ABCD are A(, ) and B(4, ). Find: (a) the gradient of DC, (b) the gradient of BC. A(, ), B(, 6) and C(7, 4) are the three vertices of a triangle. (a) Show that ABC is a right-angled isosceles triangle. (b) D is the point (5, 0). Show that BD is perpendicular to AC. (c) Eplain wh ABCD is a square. (d) Find the area of ABCD..6 The = m + c form of the equation of a straight line Consider an straight line which crosses the -ais at the point A(0, c). We sa that c is the -intercept. Let P(, ) be an other point on the line. If the gradient of the line is m then c m 0 so c m or m c. A(0, c) m 0 P(, ) c m c is the equation of a straight line with gradient m and -intercept c. a b c 0 is the general equation of a line. It has gradient a b and -intercept c. b

12 B_Chap0_08-05.qd 5/6/04 0:4 am Page 9 C: Coordinate geometr of straight lines 9 Worked eample.8 A straight line has gradient and crosses the -ais at the point (0, 4). Write down the equation of the line. The line crosses the -ais at (0, 4) so the -intercept is 4, so c 4. The gradient of the line is, so m. Using m c ou get (4). The equation of the line is 4. Worked eample.9 The general equation of a straight line is A B C 0. Find the gradient of the line, and the -intercept. 4 4 You need to rearrange the equation A B C 0 into the form m c. You can write A B C 0 as B A C or B A B C. Compare with m c, ou see that m B A and c B C, so A B C 0 is the equation of a line with gradient B A and -intercept B C. EXERCISE E Find, in the form a b c 0, the equation of the line which has: (a) gradient and -intercept, (b) gradient and -intercept, (c) gradient and -intercept. Find the gradient and -intercept for the line with equation: (a), (b) 4 5, (c) 4 7, (d) 8, (e) , (f) 0.5 4, (g) 5, (h) 4, (i).5 5, (j) 4.

13 B_Chap0_08-05.qd 5/6/04 0:4 am Page C: Coordinate geometr of straight lines.7 The = m( ) form of the equation of a straight line Consider an line which passes through the known point A(, ) and let P(, ) be an other point on the line. P(, ) If m is the gradient of the line AP, then m or m( ). A(, ) The equation of the straight line which passes through the point (, ) and has gradient m is m( ). Worked eample.0 Find the equation of the straight line which is parallel to the line 4 and passes through the point (, ). The gradient of the line 4 is 4, so the gradient of an line parallel to 4 is also 4. We need the line with gradient 4 and through the point (, ), so its equation is, using m( ), 4( ) or 4 0. Compare 4 with m c Parallel lines have equal gradients. Worked eample. (a) Find a Cartesian equation for the perpendicular bisector of the line joining A(, ) and B(4, 5). (b) This perpendicular bisector cuts the coordinate aes at C and D. Show that CD.5 AB. A(, ) 4 D C 5 B(4, 5)

14 B_Chap0_08-05.qd 5/6/04 0:4 am Page 4 C: Coordinate geometr of straight lines 4 First, we draw a rough sketch. (a) The gradient of AB 5, 4 so the gradient of the perpendicular is. The mid-point of AB is 4, (5) (, ). The perpendicular bisector is a straight line which passes through the point (, ) and has gradient, so its equation is () ( ) or. m m Using,. Using m( ). (b) Let C be the point where the line cuts the -ais. When 0, 0, so we have C(0, ). Let D be the point where the line cuts the -ais. When 0, 0 9, so we have D(9, 0) Distance CD (9 0) (0 ()) Distance AB (4 ) (5 ) C D AB so CD.5 AB An point on the -ais has -coordinate 0. An point on the -ais has -coordinate 0. Using ( ) ( ). Using a b a. b See section.4. EXERCISE F Find an equation for the straight line with gradient and which passes through the point (, 6). Find a Cartesian equation for the straight line which has gradient and which passes through (6, 0). Find an equation of the straight line passing through (, ) which is parallel to the line with equation 4. 4 Find an equation of the straight line that is parallel to 4 0 and which passes through (, ). Because a straight line equation can be arranged in a variet of was ou are usuall asked to find an equation rather than the equation. All correct equivalents would score full marks unless ou have been asked specificall for a particular form of the equation.

15 B_Chap0_08-05.qd 5/6/04 0:4 am Page 4 4 C: Coordinate geometr of straight lines 5 Find an equation of the straight line which passes through the origin and is perpendicular to the line. 6 Find the -intercept of the straight line which passes through the point (, ) and is perpendicular to the line. 7 Find a Cartesian equation for the perpendicular bisector of the line joining A(, ) and B(0, 6). 8 The verte, A, of a rectangle ABCD, has coordinates (, ). The equation of BC is. Find, in the form m c, the equation of: (a) AD, (b) AB. 9 Given the points A(0, ), B(5, 4), C(4, ) and E(, ): (a) show that BE is the perpendicular bisector of AC, (b) find the coordinates of the point D so that ABCD is a rhombus, (c) find an equation for the straight line through D and A. Hint for (b). The diagonals of a rhombus are perpendicular and bisect each other. 0 The perpendicular bisector of the line joining A(0, ) and C(4, 7) intersects the -ais at B and the -ais at D. Find the area of the quadrilateral ABCD. Show that the equation of an line parallel to a b c 0 is of the form a b k 0..8 The equation of a straight line passing through two given points The equation of the straight line which passes through the points (, ) and (, ) is B(, ) P(, ) The derivation of this equation is similar to previous ones and is left as an eercise to the reader. A(, ) Worked eample. Find a Cartesian equation of the straight line which passes through the points (, ) and (, 0).

16 B_Chap0_08-05.qd 5/6/04 0:4 am Page 4 C: Coordinate geometr of straight lines 4 If ou take (, ) (, ) and (, ) = (, 0) and substitute into the general equation, ou get, 0 or which leads to. Note that ou can check that our line passes through the points (, ) and (, 0) b seeing if the points satisf the equation. Checking for (, ) LHS RHS Checking for (, 0) LHS 0 RHS 0.9 The coordinates of the point of intersection of two lines In this section ou will need to solve simultaneous equations. If ou need further practice, see section. of chapter. 4 A point lies on a line if the coordinates of the point satisf the equation of the line. When two lines intersect, the point of intersection lies on both lines. The coordinates of the point of intersection must satisf the equations of both lines. The equations of the lines must be satisfied simultaneousl. Given accuratel drawn graphs of the two intersecting straight lines with equations a b c 0 and A B C 0, the coordinates of the point of intersection can be read off. These coordinates give the solution of the simultaneous equations a b c 0 and A B C (, ) 4 0 From the diagram, ou can clearl read off (, ) as the point of intersection of the lines 0 and 4 0. So the solution to the simultaneous equations 0 and 4 0 is,. To find the coordinates of the point of intersection of the two lines with equations a b c 0 and A B C 0, ou solve the two linear equations simultaneousl. (, )

17 B_Chap0_08-05.qd 5/6/04 0:4 am Page C: Coordinate geometr of straight lines Worked eample. Find the coordinates of the point of intersection of the straight lines with equations and 4. P At the point of intersection P, and 4. So eliminating gives 4. Rearranging gives 4. The -coordinate of P is. To find the -coordinate of P, put into The point of intersection is (, ). Worked eample.4 The straight lines with equations 5 7 and 7 intersect at the point R. Find the coordinates of R. To find the point of intersection R, ou need to solve the simultaneous equations 5 7 [A] 7 [B] 5 49 [C] 9 9 [D] Substituting in [A] gives 0 7. The coordinates of the point R are (, ). Hint. Checking that the coordinates satisf each line equation is advisable, especiall if the result is being used in later parts of an eamination question. It can usuall be done in our head rather than on paper. Checking that (, ) lies on the line ; LHS RHS Checking that (, ) lies on the line 4 ; LHS RHS 4 R Multipl equation [A] b 7 and equation [B] b. Adding [C] [D]. Checking in [A] LHS0()7RHS ; and in [B] LHS6 7()

18 B_Chap0_08-05.qd 5/6/04 0:4 am Page 45 C: Coordinate geometr of straight lines 45 EXERCISE G Verif that (, 5) lies on the line with equation. Which of the following points lie on the line with equation 6: (a) (, 0), (b) (, 0), (c) (4, ) (d) (, 6), (e) (0, )? Find the coordinates of the points where the following lines intersect the -ais: (a) 4, (b) 6, (c) 6 0, (d) The point (k, k) lies on the line with equation 6 0. Find the value of k. 5 Show that the point (4, 8) lies on the line passing through the points (, ) and (7, ). 6 (a) Find the equation of the line AB where A is the point (, 7) and B is the point (5, ). (b) The point (k, ) lies on the line AB. Find the value of the constant k. 7 A(5, ), B(, ), C(, ) and D(4, ) are the vertices of the quadrilateral ABCD. (a) Find the equation of the diagonal BD. (b) Determine whether or not the mid-point of AC lies on the diagonal BD. 8 Find the coordinates of the point of intersection of these pairs of straight lines: (a) 7 and, (b) 7 and, (c) 5 6 and 8, (d) 8 and 40, (e) 7 and 5, (f) and 5 9, (g) and 5 6, (h) 8 and Point A has coordinates,, point B has coordinates, 6 60 and point C has coordinates 9, 6 5. The straight line AC has equation 65 0 and the straight line BC has equation Write down the solution of the simultaneous equations 65 0 and

21 B_Chap0_08-05.qd 5/6/04 0:4 am Page C: Coordinate geometr of straight lines The line AB has equation 5 7. The point A has coordinates (, ) and the point B has coordinates (, k). (a) (i) Find the value of k. (ii) Find the gradient of AB. (b) Find an equation for the line through A which is perpendicular to AB. (c) The point C has coordinates (6, ). Show that AC has length p, stating the value of p. [A] The point P has coordinates (, 0) and the point Q has coordinates (4, 4). (a) Show that the length of PQ is 5. (b) (i) Find the equation of the perpendicular bisector of PQ. (ii) This perpendicular bisector intersects the -ais at the point A. Find the coordinates of A. 4 The point A has coordinates (, 5) and the point B has coordinates (, ). (a) (i) Find the gradient of AB. (ii) Show that the equation of the line AB can be written in the form r s, where r and s are positive integers. (b) The mid-point of AB is M and the line MC is perpendicular to AB. (i) Find the coordinates of M. (ii) Find the gradient of the line MC. (iii) Given that C has coordinates (5, p), find the value of the constant p. [A] 5 The points A and B have coordinates (, 5) and (9, ), respectivel. (a) (i) Find the gradient of AB. (ii) Find an equation for the line AB. (b) The point C has coordinates (, ) and the point X lies on AB so that XC is perpendicular to AB. (i) Show that the equation of the line XC can be written in the form 4 7. (ii) Calculate the coordinates of X. [A] 6 The equation of the line AB is 5 6. (a) Find the gradient of AB. (b) The point A has coordinates (4, ) and a point C has coordinates (6, 4). (i) Prove that AC is perpendicular to AB. (ii) Find an equation for the line AC, epressing our answer in the form p q r, where p, q and r are integers. (c) The line with equation also passes through the point B. Find the coordinates of B. [A]

22 B_Chap0_08-05.qd 5/6/04 0:4 am Page 49 C: Coordinate geometr of straight lines 49 7 The points A, B and C have coordinates (, 7), (5, 5) and (7, 9), respectivel. (a) Show that AB and BC are perpendicular. (b) Find an equation for the line BC. (c) The equation of the line AC is 0 and M is the mid-point of AB. (i) Find an equation of the line through M parallel to AC. (ii) This line intersects BC at the point T. Find the coordinates of T. [A] 8 The point A has coordinates (, 5), B is the point (5, ) and is the origin. (a) Find, in the form m c, the equation of the perpendicular bisector of the line segment AB. (b) This perpendicular bisector cuts the -ais at P and the -ais at Q. (i) Show that the line segment BP is parallel to the -ais. (ii) Find the area of triangle PQ. 9 The points A(, ) and C(5, ) are opposite vertices of a parallelogram ABCD. The verte B lies on the line 5. The side AB is parallel to the line 4 8. Find: (a) the equation of the side AB, (b) the coordinates of B, (c) the equations of the sides AD and CD, (d) the coordinates of D. 0 ABCD is a rectangle in which the coordinates of A and C are (0, 4) and (, ), respectivel, and the gradient of the side AB is 5. (a) Find the equations of the sides AB and BC. (b) Show that the coordinates of B is (, ). (c) Calculate the area of the rectangle. (d) Find the coordinates of the point on the -ais which is equidistant from points A and D. [A]

23 B_Chap0_08-05.qd 5/6/04 0:4 am Page C: Coordinate geometr of straight lines Ke point summar The distance between the points (, ) and (, ) p is ( ) ( ). The coordinates of the mid-point of the line segment p joining (, ) and (, ) are,. The gradient of a line joining the two points A(, ) p5 and B(, ). 4 Lines with gradients m and m : p7 are parallel if m m, are perpendicular if m m. 5 m c is the equation of a straight line with p8 gradient m and -intercept c. 6 a b c 0 is the general equation of a line. p8 It has gradient a b and -intercept c. b 7 The equation of the straight line which passes p40 through the point (, ) and has gradient m is m( ). 8 The equation of the straight line which passes p4 through the points (, ) and (, ) is. 9 A point lies on a line if the coordinates of the point p4 satisf the equation of the line. 0 Given accuratel drawn graphs of the two intersecting p4 straight lines with equations a b c 0 and A B C 0, the coordinates of the point of intersection can be read off. These coordinates give the solution of the simultaneous equations a b c 0 and A B C 0. To find the coordinates of the point of intersection of p4 the two lines with equations a b c 0 and A B C 0, ou solve the two equations simultaneousl.

24 B_Chap0_08-05.qd 5/6/04 0:4 am Page 5 C: Coordinate geometr of straight lines 5 Test ourself What to review Calculate the distance between the points (, ) and (7, 9). Section. State the coordinates of the mid-point of the line segment PQ Section. where P(, ) and Q(7, ). Find the gradient of the line joining the points A and B Section.4 where A is the point (, ) and B is the point (5, 4). 4 The lines CD and EF are perpendicular with points C(, ), Section.5 D(, 4), E(, 5) and F(k, 4). Find the value of the constant k. 5 Find a Cartesian equation of the line which passes through Sections.5 and.7 the point (, ) and is perpendicular to the line Find the point of intersection of the lines with equations Section.9 5 and 4 9. Test ourself ANSWERS. (5, ).. 4 k (, ).

8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight

### 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

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

### Chapter 3A - Rectangular Coordinate System

- Chapter A Chapter A - Rectangular Coordinate Sstem Introduction: Rectangular Coordinate Sstem Although the use of rectangular coordinates in such geometric applications as surveing and planning has been

### 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

### SLOPES AND EQUATIONS OF LINES CHAPTER

CHAPTER 90 8 CHAPTER TABLE OF CONTENTS 8- The Slope of a Line 8- The Equation of a Line 8-3 Midpoint of a Line Segment 8-4 The Slopes of Perpendicular Lines 8-5 Coordinate Proof 8-6 Concurrence of the

### REVIEW OF ANALYTIC GEOMETRY

REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.

### STRAIGHT LINE COORDINATE GEOMETRY

STRAIGHT LINE COORDINATE GEOMETRY (EXAM QUESTIONS) 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

### Chapter 12. The Straight Line

302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic- geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,

### The Graph of a Linear Equation

4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

### 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

### DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

### 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

### 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?

### 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

0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral

### 13 Graphs, Equations and Inequalities

13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,

### STRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2

STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan θ is called the slope

### 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

### Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304

Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straight-line depreciation. The Circle Definition Anone who has drawn a circle using a compass

.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

### 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

### SLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT

. Slope of a Line (-) 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail

### 3 Rectangular Coordinate System and Graphs

060_CH03_13-154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter

### LINEAR FUNCTIONS OF 2 VARIABLES

CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for

### CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES

6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits

### 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

### INTEGRATION FINDING AREAS

INTEGRTIN FINDING RES Created b T. Madas Question 1 (**) = 4 + 10 3 The figure above shows the curve with equation = 4 + 10, R. Find the area of the region, bounded b the curve the coordinate aes and the

### 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

### INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

### Coordinate Coplanar Distance Formula Midpoint Formula

G.(2) Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the oneand two-dimensional coordinate systems to

### 2.6. The Circle. Introduction. Prerequisites. Learning Outcomes

The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle - in particular

### P1. Plot the following points on the real. P2. Determine which of the following are solutions

Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in

### SL Calculus Practice Problems

Alei - Desert Academ SL Calculus Practice Problems. The point P (, ) lies on the graph of the curve of = sin ( ). Find the gradient of the tangent to the curve at P. Working:... (Total marks). The diagram

### Winter 2016 Math 213 Final Exam. Points Possible. Subtotal 100. Total 100

Winter 2016 Math 213 Final Exam Name Instructions: Show ALL work. Simplify wherever possible. Clearly indicate your final answer. Problem Number Points Possible Score 1 25 2 25 3 25 4 25 Subtotal 100 Extra

### STRAND: ALGEBRA Unit 3 Solving Equations

CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

### 8 Graphs of Quadratic Expressions: The Parabola

8 Graphs of Quadratic Epressions: The Parabola In Topic 6 we saw that the graph of a linear function such as = 2 + 1 was a straight line. The graph of a function which is not linear therefore cannot be

### TRANSFORMATIONS AND THE COORDINATE PLANE

CHPTER TRNSFORMTIONS ND THE COORDINTE PLNE In our stud of mathematics, we are accustomed to representing an equation as a line or a curve. This blending of geometr and algebra was not alwas familiar to

A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which

### SECTION 2.2. Distance and Midpoint Formulas; Circles

SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation

### Graphing and transforming functions

Chapter 5 Graphing and transforming functions Contents: A B C D Families of functions Transformations of graphs Simple rational functions Further graphical transformations Review set 5A Review set 5B 6

### 2.6. The Circle. Introduction. Prerequisites. Learning Outcomes

The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures. In this brief Section we discuss the basic coordinate geometr of a circle - in particular the basic equation representing

### 2.1 Equations of Lines

Section 2.1 Equations of Lines 1 2.1 Equations of Lines The Slope-Intercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is

### The Cartesian Plane The Cartesian Plane. Performance Criteria 3. Pre-Test 5. Coordinates 7. Graphs of linear functions 9. The gradient of a line 13

6 The Cartesian Plane The Cartesian Plane Performance Criteria 3 Pre-Test 5 Coordinates 7 Graphs of linear functions 9 The gradient of a line 13 Linear equations 19 Empirical Data 24 Lines of best fit

### 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,

### PROPERTIES OF TRIANGLES AND QUADRILATERALS

Mathematics Revision Guides Properties of Triangles, Quadrilaterals and Polygons Page 1 of 21 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier PROPERTIES OF TRIANGLES AND QUADRILATERALS

### 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

### 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

### 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,

### HI-RES STILL TO BE SUPPLIED

1 MRE GRAPHS AND EQUATINS HI-RES STILL T BE SUPPLIED Different-shaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object

### CHAPTER 8 QUADRILATERALS. 8.1 Introduction

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

### 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

### Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

### Higher Geometry Problems

Higher Geometry Problems ( Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement

### If (a)(b) 5 0, then a 5 0 or b 5 0.

chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic

### Quadratic Functions. MathsStart. Topic 3

MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE

### 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

### ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude

ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height

1. Determine whether each quadrilateral is a Justify your answer. 3. KITES Charmaine is building the kite shown below. She wants to be sure that the string around her frame forms a parallelogram before

### 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

### 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

### Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()

### Circle theorems CHAPTER 19

352 CHTE 19 Circle theorems In this chapter you will learn how to: use correct vocabulary associated with circles use tangent properties to solve problems prove and use various theorems about angle properties

### 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

### Practice Unit tests Use this booklet to help you prepare for all unit tests in Higher Maths.

Practice Unit tests Use this booklet to help you prepare for all unit tests in Higher Maths. Your formal test will be of a similar standard. Read the description of each assessment standard carefully to

### 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

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

### 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

### 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

### 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

### 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

### 4 BASIC GEOMETRICAL IDEAS

4 BASIC GEOMETRICAL IDEAS Q.1. Use the figure to name. (a) Five points (b) A line (c) Four rays (d) Five line segments Ans. (a) O, B, C, D and E. (b) DB, OB etc. (c) OB, OC, OD and ED Exercise 4.1 (d)

### 3. Lengths and areas associated with the circle including such questions as: (i) What happens to the circumference if the radius length is doubled?

1.06 Circle Connections Plan The first two pages of this document show a suggested sequence of teaching to emphasise the connections between synthetic geometry, co-ordinate geometry (which connects algebra

### 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

### 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

### AQA Level 2 Certificate FURTHER MATHEMATICS

AQA Qualifications AQA Level 2 Certificate FURTHER MATHEMATICS Level 2 (8360) Our specification is published on our website (www.aqa.org.uk). We will let centres know in writing about any changes to the

### 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

### 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

### 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

### Transformations Packet Geometry

Transformations Packet Geometry 1 Name: Geometry Chapter 14: Transformations Date Due Section Topics Assignment 14.1 14.2 14.3 Preimage Image Isometry Mapping Reflections Note: Notation is different in

CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent

### 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

### 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

### 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

### 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

### 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

### 2 Analysis of Graphs of

ch.pgs1-16 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,

### Lesson 6: Linear Functions and their Slope

Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation

### Alex and Morgan were asked to graph the equation y = 2x + 1

Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged

### 1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered

Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

### 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

### 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,

### NATIONAL QUALIFICATIONS

H Mathematics Higher Paper 1 Practice Paper A Time allowed 1 hour 0 minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions 1 0 (40 marks) Instructions

### 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,

### Circles - Past Edexcel Exam Questions

ircles - Past Edecel Eam Questions 1. The points A and B have coordinates (5,-1) and (13,11) respectivel. (a) find the coordinates of the mid-point of AB. [2] Given that AB is a diameter of the circle,

### Chapter 8. Lines and Planes. By the end of this chapter, you will

Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes

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

### Archimedes quadrature of the parabola and the method of exhaustion

JOHN OTT COLLEGE rchimedes quadrature of the parabola and the method of ehaustion CLCULUS II SCIENCE) Carefull stud the tet below and attempt the eercises at the end. You will be evaluated on this material