122. Show that the following points, taken in order, form the figure mentioned against the points.
|
|
- Magdalen Warren
- 7 years ago
- Views:
Transcription
1 Find the distances between the following pair of points 117. (i) (b + c, c + a) and (c + a, a + b) (ii) (a cos, a sin) and (a cos, a sin) (iii) (am 1, am 1 ) and (am, am ) (i) Find the value of x 1 if the distance between the points (x 1, ) and (3, 4) be 8. (ii) Prove that the points (a, 4a), (a, 6a) and (a + 3a,5a) are the vertices of an equilateral triangle whose side is a. (iii) Prove that the points (, 1), (1, 0), (4, 3) and (1, ) are at the vertices of a parallelogram Prove that the point, is the centre of the circle circumscribing the triangle whose angular points are (1, ), (, 3) and (, ). 10. Find the coordinates of the points which (i) Divides the line joining the points (1, 3) and (, 7) in the ratio 3 : 4. (ii) Divides, internally and externally, the line joining ( 1, ) to (4, 5) in the ratio : (i) The line joining the points (1, ) and ( 3, 4) is trisected; find the coordinates of the points of trisection. (ii) Find the points which divide the line segment joining (8, 1) and (1, 8) into four equal parts. (iii) If (, 1), (1, 0) and (4, 3) are three successive vertices of parallelogram, find the fourth vertex. 1. Show that the following points, taken in order, form the figure mentioned against the points. (i) (, 5), (3, 4), (7, 10)... Right angled isosceles triangle. (ii) (1, 3), (3, 1), ( 5, 5)... Right angled triangle. (iii) ( 3, 1), ( 6, 7), (3, 9), (6, 1)... Parallelogram. (iv) (8, 4), (5, 7), ( 1, 1), (, )... Rectangle. (v) (3, ), (7, 6), ( 1, ), ( 5, 6)... Rhombus. 13. Find the area of the triangle having vertices (i) (0, 4), (3, 6) and ( 8, ). (ii) (5, ), ( 9, 3) and ( 3, 5). (iii) (a, c + a), (a, c) and ( a, c a). (iv) (a cos 1, b sin 1 ), (a cos, b sin ) and (a cos 3, b sin 3 ). (v) (am 1, am 1 ), (am, am ) and (am 3, am 3 ). (vi) {am 1 m, a(m 1 + m )}, {am m 3, a(m + m 3 )} and (am 3 m 1, a(m 3 + m 1 )} a a a (vii) am 1,, am, and am 3,. m1 m m3 14. Prove that the following sets of three points are in a straight line: (i) (1, 4), (3, ), and ( 3, 16). 1 (ii),3,( 5,6 ) and ( 8, 8). (iii) (a, b + c), (b, c + a), and (c, a + b) 15. Find the area of the quadrilaterals the coordinates of whose angular points, taken in order, are: (i) (1, 1), (3, 4), (5, ) and (4, 7). (ii) ( 1, 6), ( 3, 9), (5, 8) and (3, 9). 16. (i) Find the value of k, if the area of the triangle formed by (k, 0), (3, 4) and (5, ) is 10 sq. units. (ii) Show that the following points are collinear (0, ), ( 1, 1), (, 4).
2 (iii) Find the value of k if the points (k, 1), (, 1) and (4, 5) are collinear. 17. (i) Find the centroid of the triangle whose vertices are (, 7), (3, 1) and ( 5, 6). (ii) Find the In centre of the triangle, whose vertices are A(3, ), B(7, ) and C(7, 5). 18. (i) Find the coordinates of (1, ) with respect to new axes, when the origin is shifted to (, 3) by translation of axes. (ii) Find the transformed equation of x + 4xy + 5y = 0, when the origin is shifted to (3, 4) by the translation of axes. (iii) The coordinates of a point are changed as ( 4, 3), when the origin is shifted to (1, 5) by the translation of axes. Find the coordinates of the point in the original system. (iv) When the origin is shifted to (4, 5) by the translation of axes, find the coordinates of the following points with reference to new axes. (a) (0, 3) (b) (, 4) (c) (4, 5) 19. (i) Find the point to which the origin is to be shifted by translation of axes so as to remove the first degree terms from the equation ax + hxy + by + gx + fy + c = 0, where h ab. (ii) Find the point to which the origin is to be shifted by the translation of axes so as to remove the first degree terms from the equation ax + by + gx + fy + c = 0, where a 0, b 0. (iii) Find the point to which the origin is to be shifted so as to remove the first degree terms from the equation 4x + 9y 8x + 36y + 4 = (i) Find the point to which the origin is to be shifted so that the point (3, 0) may change to (, 3). (ii) Find the transformed equations of the following when the origin is shifted to ( 1, ) by translation of axes. (a) x + y + x 4y + 1 = 0 (b) x + y 4x + 4y = 0 (iii) The point to which the origin is shifted and the transformed equation are given below. Find the original equation. (a) (3, 4); x + y = 4 (b) ( 1, ); x + y + 16 = 0 (iv) If the transformed equation of a curve is x + 3xy y + 17x 17y 11 = 0, when the origin is shifted to the point (, 3), then find the original equation of the curve (i) Find the equation of the straight line making an angle of 10 with the positive direction of the X axis and passing through the point (0, ). (ii) Find the equation of the straight line which makes an angle 135 with the positive X axis and passes through the point (, 3). (iii) Write the equations of the straight lines parallel to Y axis and (i) at a distance of units from the Y axis to the right of it, (ii) at a distance of 5 units from the Y axis to the left of it. (iv) Find the value of x, if the slope of the line passing through (, 5) and (x, 3) is. 13. Find the equations of the straight lines which makes the following angles with the positive X axis in the positive direction and which pass through the points given below. (i) 4 and (0, 0) (ii) 4 and (1, ) (iii) 135 and (3, ) (iv) 150 and (, 1) 133. (i) Find the equations of the straight lines passing through the origin and making equal angles with the coordinate axes. (ii) Find the equation of the straight line passing through ( 4, 5) and cutting off equal nonzero intercepts on the coordinate axes. (iii) Find the equation of the straight line passing through (, 4) and making nonzero intercepts whose sum is zero. (iv) Find the equation of the straight line passing through the point (4, 3) and perpendicular to the line passing through the points (1, 1) and (, 3) Find the equation of the straight line passing through
3 a a (i) (a cos 1, a sin 1 ) and (a cos, a sin ) (ii) at 1, andat, t1 t (iii) (at 1, at 1 ) and (at, at ) 135. (i) Find the equation to the straight line which passes through the point ( 4, 3) and is such that the portion of it between the axes is divided by the point in the ratio 5 : 3. (ii) Find the equation to the straight lines which passes through the point (1, ) and cut off equal intercept from the axes In what follows, p denotes the distance of the straight line from the origin. The normal ray drawn from the origin to the straight line makes an angle with the positive direction of the X axis. Find the equations of the straight lines with the following values of p and. (i) p = 5, = 60 (ii) p = 6, = 150 (iii) p = 3/, = /3 (iv) p = 1, = 7/4 (v) p = 4, = 90 (vi) p =.5, = Find the equations of the straight lines in the symmetric form, in the following cases having the given slope and passing through the given point. (i) Slope = 3, point (, 3) (ii) Slope = 1, point (, 0) (iii) Slope = 1, point (1, 1) (i) Reduce to the perpendicular form the equation x + y3 + 7 = 0. (ii) A straight line parallel to the line y = 3x passes through Q(, 3) and cuts the line x + 4y 7 = 0 at P. Find the length of PQ. (iii) Find the points on the line 4x 3y 10 = 0 which are at a distance of 5 units from the point (1, ). (iv) Transform the equation 5x y 7 = 0 into (i) slope intercept form (ii) intercept form and (iii) normal form 139. (a) (i) Find the value of k, if the straight lines 6x 10y + 3 = 0 and kx 5y + 8 = 0 are parallel. (ii) Find the value of p, if the straight lines 3x + 7y 1 = 0 and 7x py + 3 = 0 are mutually perpendicular. (iii) Find the value of k, so that the straight lines y 3kx + 4 = 0 and (k 1)x (8k 1)y 6 = 0 are perpendicular. (b) (iv) The line x y 1meets the X axis at P. Find the equation of the line perpendicular to this line at P. Find the angles between the pairs of straight lines: (i) x y3 = 5 and 3x + y = 7 (ii) x 4y = 3 and 6x y = 11 (iii) y = 3x + 7 and 3y x = 8 (iv) y = ( 3)x + 5 and y = ( + 3)x 7 (v) (m mn)y = (mn + n )x + n 3 and (mn + m )y = (mn n )x + m Find the equation to the straight line passing through the point ( 6, 10) and perpendicular to the straight line 7x + 8y = Find the equation to the straight line (i) Passing through the point (, 3) and perpendicular to the straight line 4x 3y = 10. (ii) Passing through the point (, 3) and perpendicular to the straight line joining the points (5, 7) and ( 6, 3). (iii) Passing through the point ( 4, 3) and perpendicular to the straight line joining (1, 3) and (, 7)
4 14. (i) Prove that the equation to the straight line which passes through the point (a cos 3, a sin 3 ) and is perpendicular to the straight line x sec + y cosec = a is x cos y sin = a cos. (ii) Find the equations to the straight lines passing through (x, y) and respectively perpendicular to the straight lines (a) xx + yy = a xx' yy', (b) 1, 143. Find the angle between the two straight lines 3x = 4y + 7 and 5y = 1x + 6 and also the equations to the two straight lines which pass through the point (4, 5) and make equal angles with the two given lines (i) Find the equations to the straight lines which pass through the origin and are inclined at 75 to the straight line x + y + 3 (y x) = a (ii) Find the equations to the straight lines which pass through the point (h, k) and are inclined at an angle tan 1 m to the straight line y = mx + c (i) Find the values of k, if the angle between the straight lines kx + y + 9 = 0 and y 3x = 4 is 45 (ii) Find the equations of the straight lines passing through the point ( 10, 4) and making an angle with the line x y = 10 such that tan = Find the equations of the straight lines passing through the point (1, ) and making an angle of 60 with the line 3x + y = Find the length of the perpendicular drawn from (i) The point (4, 5) upon the straight line 3x + 4y = 10. (ii) The origin upon the straight line x y (iii) The point ( 3, 4) upon the straight line 1(x + 6) = 5(y ). (iv) The point (b, a) upon the straight line x y 1. (v) Find the length of the perpendicular from the origin upon the straight line joining the two points whose coordinates are (a cos, a sin ) and (a cos, a sin ) 148. Show that the product of the perpendiculars drawn from the two points ( a b, 0) upon the straight line cos sin 1is b (i) Find the distance between the two parallel straight lines y = mx + c and y = mx + d (ii) What are the points on the axis of x whose perpendicular distance from the straight line 1 is a? 150. (i) Find the distance between parallel straight lines 3x + 4y 3 = 0 and 6x + 8y 1 = 0. (ii) Find the distance between the following parallel lines: (a) 3x 4y = 1, 3x 4y = 7 (b) 5x 3y 4 = 0, 10x 6y 9 = Find the orthocentre of the triangle whose vertices are ( 5, 7), (13, ) and ( 5, 6). 15. (i) Find the length of the perpendicular drawn from the point given against the following straight lines. (a) 5x y + 4 = 0... (, 3) (b) 3x 4y + 10 = 0... (3, 4) (c) x 3y 4 = 0... (0, 0) (ii) Find the perpendicular distance from the point ( 6, 5) to the straight lines 5x 1y = (i) Find the foot of the perpendicular drawn from (4, 1) upon the straight line 3x 4y + 1 = 0 (ii) Find the foot of the perpendicular drawn from ( 1, 3) on the line 5x y = 18.
5 (iii) Find the foot of the perpendicular drawn from (3, 0) upon the straight line 5x + 1y 14 = (i) Find the image of the point (1, ) w.r.t the straight line 3x + 4y 1 = 0 (ii) Find the image of (, 3) w.r.t the straight line 4x 5y + 8 = 0 (iii) x 3y 5 = 0 is the perpendicular bisector of the line segment joining the points A, B. If A = ( 1, 3), find the coordinates of B. (iv) Prove that the feet of the perpendiculars from the origin on the lines x + y = 4, x + 5y = 6 and 15x 7y = 44 all lie on a straight line Show that the distance of the point (6, ) from the line 4x + 3y = 1 is half the distance of the point (3, 4) from the line 4x 3y = Each side of a square is of length 4 units. The centre of the square is (3, 7) and one of its diagonals is parallel to y = x. Find the coordinates of its vertices Find the equations to the straight lines bisecting the angles between the following pairs of straight lines, placing first the bisector of the angle in which the origin lies. (i) 1x + 5y 4 = 0 and 3x + 4y + 7 = 0. (ii) 4x + 3y 7 = 0 and 4x + 7y 31 = 0. (iii) x + y = 4 and y + 3x = Find the equations to the bisectors of the internal angles of the triangles the equations of whose sides are respectively (i) 3x + 4y = 6, 1x 5y = 3, and 4x 3y + 1 = 0. (ii) 3x + 5y = 15, x + y = 4, and x + y = (i) Find the direction in which a straight line must be drawn through the point (1, ) so that its point of intersection with the line x + y = 4 may be at a distance of 1 6 from this point. 3 (ii) Find the equation of the straight line passing through the point of intersection of the lines x + y + 1 = 0 and x y + 5 = 0 and containing the point (5, ) (i) Find the value of k, if the lines x 3y + k = 0, 3x 4y 13 = 0 and 8x 11y 33 = 0 are concurrent. (iii) If the straight lines ax + by + c = 0, bx + cy + a = 0 and cx + ay + b = 0 are concurrent, then prove that a 3 + b 3 + c 3 = 3abc. (iv) Show that the lines x + y 3 = 0, 3x + y = 0 and x 3y 3 = 0 are concurrent and find the point of concurrency. (v) Show that the straight line (a b)x + (b c)y = c a, (b c)x + (c a)y = a b and (c a)x + (a b)y = b c are concurrent (i) Find the value of p, if the following lines are concurrent. (a) 3x + 4y = 5, x + 3y = 4, px + 4y = 6 (b) 4x 3y 7 = 0, x + py + = 0, 6x + 5y 1 = 0 (ii) If 3a + b + c = 0, then show that the equation ax + by + c = 0 represents a family of concurrent straight lines and find the point of concurrency. 16. (i) Show that the straight lines x + y = 0, 3x + y 4 = 0 and x + 3y 4 = 0 form an isosceles triangle. (ii) Find the area of the triangle formed by the straight lines x y 5 = 0, x 5y + 11 = 0 and x + y 1 = (i) Find the locus of the third vertex of a right angled triangle, the ends of whose hypotenuse are (4, 0) and (0, 4). (ii) Find the equation of the locus of a point which is equidistant from the points A( 3, ) and B(0, 4).
6 164. (i) Find the equation of the locus of point P such that the distance of P from the origin is twice the distance of P from A(1, ). (ii) A(, 3) and B( 3, 4) be two given points. Find the equation of the locus of P so that the area of the triangle PAB is 8.5 sq. units (i) Find the equation of locus of a point P, if the distance of P from A (3, 0) is twice the distance of P from B( 3, 0). (ii) Find the equation of locus of a point which is equidistant from the coordinate axes (i) Find the equation of the locus of a point P, the square of whose distance from the origin is 4 times its y coordinate. (ii) Find the equation of locus of a point equidistant from A (, 0) and the Y axis (i) Find the equation of locus of P, if the ratio of the distances from P to (5, 4) and (7, 6) is : 3.
7 (ii)find the equation of locus of P, if the line segment joining (, 3) and ( 1, 5) subtends a right angle at P The ends of the hypotenuse of a right angled triangle are (0, 6) and (6, 0). Find the equation of locus of its third vertex (i) Find the equation of locus of a point the difference of whose distance from ( 5, 0) and (5, 0) is 8 units. (ii) Find the equation of locus of P, if A = (4, 0), B = ( 4, 0) and PA + PB = (i) Find the equation of locus of a point, the sum of whose distances from (0, ) and (0, ) is 6 units. (ii) Find the equation of locus of P, if A = (, 3), B = (, 3) and PA + PB = (i) A (5, 3) and B (3, ) are two fixed points. Find the equation of locus of P, so that the area of triangle PAB is 9 units. (ii) Find the equation of locus of the point P such that PA + PB = c, where A = (a, 0), B = ( a, 0) and 0 < a < c. 17. If the distance from P to the points (, 3) and (, 3) are in the ratio : 3, then find the equation of locus of P A (1, ), B (, 3) and C (, 3) are three points. A point P moves such that PA + PB = PC. Find the equation to the locus of P (i) (a b) (b c) (ii) asin (iii) a(m m 1) (m1 m ) (i) x 1 = (i), (ii) 1,, ( 11, 4) (i),0, (ii) (9, 11) (10, 10) (11, 9) (iii) (1, ) (i) 1 (ii) 9 (iii) a (iv) ab sin sin sin (v) a (m m 3 )(m 3 m 1 )(m 1 m ) (vi) ½ a (m 1 m )(m m 3 )(m 3 m 1 ) (vii) ½ a (m 1 m )(m m 3 )(m 3 m 1 ) + m 1 m m (i) 0½ (ii) (i) k = 1 or k = 3 3 (iii) k = (i) (0, 4) (ii) (6, 3) 18. (i) (3, 1) (ii) x + 4xy + 5y + 8x + 5y = 0 (iii) ( 3, 8) (iv) (a) ( 4, 8) (b) ( 6, 9) (c) (0, 0) hf bg gh af 19. (ii), ab h ab h (iii) (, ) (1, ) g f (ii), a b (iv) 1 7, (i) (1, 3) (ii) x + y 4 = 0 (iii) x + y + x 8y + 5 = 0 (iv) x y + 3xy + 4x 11y + 10 = (i) y = 3x (ii) y = x + 1 (iii) x =, x = 5 (iv) x = 1
8 13. (i) y = x (ii) x y + 1 = 0 (iii) x + y = 1 (iv) x + 3 y + ( + 3 ) = (i) y = x (ii) x + y = 1 (iii) x y + 6 = 0 (iv) x + y + = (i) xcos y sin acos (ii) t 1 t y + x = a(t 1 + t ) (iii) y(t 1 + t ) x = at 1 t 135. (i) 0y 9x = 96 (ii) x + y + 1 = (i) x + 3 y 10 = 0 (ii) 3 x y + 1 = 0 (iii) x 3 y + 3 = 0 (iv) x y = 0 (v) y = 4 (vi) x = 5/ (i) 3 (x ) = (y 3) (ii) y 0 = (x + ) (iii) y + x = (i) x cos40 + y sin40 = 7 1 (ii) PQ = (iii) x = 4, y = 1 3 (iv) (a) y = 5 x 7, 139. (a) (i) k = 3, (ii) p = 3, (iii) k = 1 or k = 1/6, (iv) ax + by = a (b) (i), (ii) _ 1 3 tan 10, (iii) _ 1 4 tan 4m n, (iv) 60, (v) tan 3 m n _ x 7y = (i) 3x + 4y = 18 (ii) 11x + 4y = 10 (iii) x + 4y + 16 = (ii) (a) yx xy = 0 (b) a yx b xy = (a b )(xy) tan, 7x + 9y = 17, 7y 9x + 1 = (i) y + 3 x = 0, x = 0 (ii) y = k, y k = m (x h) 1 m 145. (i) k =, 1/ (ii) x + 10 = 0, 3x + 4y + 14 = y =, y = 3 (x 1) 147. (i) 4 5, (ii) 5, (iii) , (iv) c d a 149. (i) b,0 1 m b (i) ½, (ii) (a) 1, (b) (i) (a) 0, (b) 3 5, (c) 4 10, (ii) a ab b a b 151. ( 3, ) (ii) 93 13, (v) acos
9 (i), (ii) (4, ), (iii), 154. (i), , (ii) 6 10, 15 3 (iii) 8 6, (1, 5) (1, 9) (5, 8) (5, 5) 159. (i) = 15 or 75 (ii) 3x + 7y 1 = (i) k = 7, (ii) (4, 5) 161. (i) (a), (b) 4, (ii) (3, ) 16. (ii) 9 sq. units (i) x + y 4x 4y = 0 (ii) 6x + 4y = (i) 3x + 3y 8x 16y + 0 = 0 (ii) x + 5y = 0, x + 5y = (i) x + y + 10x + 9 = 0 (ii) x y = (i) x + y 4y = 0 (ii) y 4x + 4 = (i) 5(x + y ) 34x + 10y + 9 = 0 (ii) x + y x 8y + 13 = x + y 6x 6y = (i) 1 (ii) (i) 1 (ii) 16x + 7y 64x 48 = (i) (5x y 37)(5x y 1) = 0 (ii) x + y = c a x + 5y 0x 78y + 65 = x 7y + 4 = 0
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
More informationSelected 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
More informationPOINT 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
More informationDefinitions, 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
More informationConjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true)
Mathematical Sentence - a sentence that states a fact or complete idea Open sentence contains a variable Closed sentence can be judged either true or false Truth value true/false Negation not (~) * Statement
More informationDEFINITIONS. 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
More informationConjectures for Geometry for Math 70 By I. L. Tse
Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:
More informationChapter 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
More informationChapters 6 and 7 Notes: Circles, Locus and Concurrence
Chapters 6 and 7 Notes: Circles, Locus and Concurrence 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
More informationVisualizing Triangle Centers Using Geogebra
Visualizing Triangle Centers Using Geogebra Sanjay Gulati Shri Shankaracharya Vidyalaya, Hudco, Bhilai India http://mathematicsbhilai.blogspot.com/ sanjaybhil@gmail.com ABSTRACT. In this paper, we will
More informationConjectures. 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
More informationAnalytical 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
More informationhttp://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
More informationGeometry 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
More informationGEOMETRY CONCEPT MAP. Suggested Sequence:
CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons
More informationSection 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,
More informationContents. 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..................................
More informationCIRCLE 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
More informationGeometry 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
More informationSan 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
More information39 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
More informationAlgebra Geometry Glossary. 90 angle
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
More informationCircle 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
More informationIMO 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
More informationAdvanced Euclidean Geometry
dvanced Euclidean Geometry What is the center of a triangle? ut what if the triangle is not equilateral?? Circumcenter Equally far from the vertices? P P Points are on the perpendicular bisector of a line
More informationQUADRILATERALS 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
More informationCo-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
More information1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
More informationTarget To know the properties of a rectangle
Target To know the properties of a rectangle (1) A rectangle is a 3-D shape. (2) A rectangle is the same as an oblong. (3) A rectangle is a quadrilateral. (4) Rectangles have four equal sides. (5) Rectangles
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More informationArea. Area Overview. Define: Area:
Define: Area: Area Overview Kite: Parallelogram: Rectangle: Rhombus: Square: Trapezoid: Postulates/Theorems: Every closed region has an area. If closed figures are congruent, then their areas are equal.
More informationBiggar 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
More informationThe 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
More informationCSU 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
More informationMath 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)
More informationThe 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
More information( 1 ) Obtain the equation of the circle passing through the points ( 5, - 8 ), ( - 2, 9 ) and ( 2, 1 ).
PROBLEMS 03 CIRCLE Page ( ) Obtain the equation of the irle passing through the points ( 5 8 ) ( 9 ) and ( ). [ Ans: x y 6x 48y 85 = 0 ] ( ) Find the equation of the irumsribed irle of the triangle formed
More informationSolutions to Practice Problems
Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles
More informationThe 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
More informationMathematics Notes for Class 12 chapter 10. Vector Algebra
1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is non-negative
More informationNew York State Student Learning Objective: Regents Geometry
New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students
More informationPOTENTIAL REASONS: Definition of Congruence:
Sec 6 CC Geometry Triangle Pros Name: POTENTIAL REASONS: Definition Congruence: Having the exact same size and shape and there by having the exact same measures. Definition Midpoint: The point that divides
More informationAngles 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,
More information13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant
æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the
More informationTHREE 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,
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationIMO Training 2008 Circles Yufei Zhao. Circles. Yufei Zhao.
ircles Yufei Zhao yufeiz@mit.edu 1 Warm up problems 1. Let and be two segments, and let lines and meet at X. Let the circumcircles of X and X meet again at O. Prove that triangles O and O are similar.
More informationLesson 3.1 Duplicating Segments and Angles
Lesson 3.1 Duplicating Segments and ngles In Exercises 1 3, use the segments and angles below. Q R S 1. Using only a compass and straightedge, duplicate each segment and angle. There is an arc in each
More informationUnit 2 - Triangles. Equilateral Triangles
Equilateral Triangles Unit 2 - Triangles Equilateral Triangles Overview: Objective: In this activity participants discover properties of equilateral triangles using properties of symmetry. TExES Mathematics
More informationParametric Equations and the Parabola (Extension 1)
Parametric Equations and the Parabola (Extension 1) Parametric Equations Parametric equations are a set of equations in terms of a parameter that represent a relation. Each value of the parameter, when
More information/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
More informationGEOMETRY COMMON CORE STANDARDS
1st Nine Weeks Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,
More information2006 Geometry Form A Page 1
2006 Geometry Form Page 1 1. he hypotenuse of a right triangle is 12" long, and one of the acute angles measures 30 degrees. he length of the shorter leg must be: () 4 3 inches () 6 3 inches () 5 inches
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More informationChapter 8 Geometry We will discuss following concepts in this chapter.
Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles
More informationCHAPTER 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
More informationThe 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
More informationExercise 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
More informationGEOMETRIC 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
More informationEquation 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
More informationPractice Book. Practice. Practice Book
Grade 10 Grade 10 Grade 10 Grade 10 Grade 10 Grade 10 Grade 10 Exam CAPS Grade 10 MATHEMATICS PRACTICE TEST ONE Marks: 50 1. Fred reads at 300 words per minute. The book he is reading has an average of
More informationGeometry Enduring Understandings Students will understand 1. that all circles are similar.
High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,
More informationBALTIC OLYMPIAD IN INFORMATICS Stockholm, April 18-22, 2009 Page 1 of?? ENG rectangle. Rectangle
Page 1 of?? ENG rectangle Rectangle Spoiler Solution of SQUARE For start, let s solve a similar looking easier task: find the area of the largest square. All we have to do is pick two points A and B and
More information56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points.
6.1.1 Review: Semester Review Study Sheet Geometry Core Sem 2 (S2495808) Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which
More informationGEOMETRY. Constructions OBJECTIVE #: G.CO.12
GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic
More informationAlgebra 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?
More informationChapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.
Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.
More informationThe 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
More informationSemester Exam Review. Multiple Choice Identify the choice that best completes the statement or answers the question.
Semester Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Are O, N, and P collinear? If so, name the line on which they lie. O N M P a. No,
More informationPractical 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
More informationTriangle Centers MOP 2007, Black Group
Triangle Centers MOP 2007, Black Group Zachary Abel June 21, 2007 1 A Few Useful Centers 1.1 Symmedian / Lemmoine Point The Symmedian point K is defined as the isogonal conjugate of the centroid G. Problem
More information" Angles ABCand DEFare congruent
Collinear points a) determine a plane d) are vertices of a triangle b) are points of a circle c) are coplanar 2. Different angles that share a common vertex point cannot a) share a common angle side! b)
More informationSIMSON S THEOREM MARY RIEGEL
SIMSON S THEOREM MARY RIEGEL Abstract. This paper is a presentation and discussion of several proofs of Simson s Theorem. Simson s Theorem is a statement about a specific type of line as related to a given
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationGeometry. Higher Mathematics Courses 69. Geometry
The fundamental purpose of the course is to formalize and extend students geometric experiences from the middle grades. This course includes standards from the conceptual categories of and Statistics and
More informationHigh School Geometry Test Sampler Math Common Core Sampler Test
High School Geometry Test Sampler Math Common Core Sampler Test Our High School Geometry sampler covers the twenty most common questions that we see targeted for this level. For complete tests and break
More informationGeometry Course Summary Department: Math. Semester 1
Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give
More informationCumulative Test. 161 Holt Geometry. Name Date Class
Choose the best answer. 1. P, W, and K are collinear, and W is between P and K. PW 10x, WK 2x 7, and PW WK 6x 11. What is PK? A 2 C 90 B 6 D 11 2. RM bisects VRQ. If mmrq 2, what is mvrm? F 41 H 9 G 2
More informationCHAPTER 1. LINES AND PLANES IN SPACE
CHAPTER 1. LINES AND PLANES IN SPACE 1. Angles and distances between skew lines 1.1. Given cube ABCDA 1 B 1 C 1 D 1 with side a. Find the angle and the distance between lines A 1 B and AC 1. 1.2. Given
More informationThe 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
More information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
More informationLesson 5-3: Concurrent Lines, Medians and Altitudes
Playing with bisectors Yesterday we learned some properties of perpendicular bisectors of the sides of triangles, and of triangle angle bisectors. Today we are going to use those skills to construct special
More informationGeoGebra. 10 lessons. Gerrit Stols
GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter
More informationThe 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
More informationStraight 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
More informationThe Geometry of Piles of Salt Thinking Deeply About Simple Things
The Geometry of Piles of Salt Thinking Deeply About Simple Things PCMI SSTP Tuesday, July 15 th, 2008 By Troy Jones Willowcreek Middle School Important Terms (the word line may be replaced by the word
More informationGeometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
More informationGeorgia Online Formative Assessment Resource (GOFAR) AG geometry domain
AG geometry domain Name: Date: Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (1 of 36 ) 1. Amy drew a circle graph to represent
More informationChapter 1. The Medial Triangle
Chapter 1. The Medial Triangle 2 The triangle formed by joining the midpoints of the sides of a given triangle is called the medial triangle. Let A 1 B 1 C 1 be the medial triangle of the triangle ABC
More informationSandia High School Geometry Second Semester FINAL EXAM. Mark the letter to the single, correct (or most accurate) answer to each problem.
Sandia High School Geometry Second Semester FINL EXM Name: Mark the letter to the single, correct (or most accurate) answer to each problem.. What is the value of in the triangle on the right?.. 6. D.
More informationWarm-up Theorems about triangles. Geometry. Theorems about triangles. Misha Lavrov. ARML Practice 12/15/2013
ARML Practice 12/15/2013 Problem Solution Warm-up problem Lunes of Hippocrates In the diagram below, the blue triangle is a right triangle with side lengths 3, 4, and 5. What is the total area of the green
More information5.1 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
More informationMost popular response to
Class #33 Most popular response to What did the students want to prove? The angle bisectors of a square meet at a point. A square is a convex quadrilateral in which all sides are congruent and all angles
More informationWarm-up Tangent circles Angles inside circles Power of a point. Geometry. Circles. Misha Lavrov. ARML Practice 12/08/2013
Circles ARML Practice 12/08/2013 Solutions Warm-up problems 1 A circular arc with radius 1 inch is rocking back and forth on a flat table. Describe the path traced out by the tip. 2 A circle of radius
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More information11.3 Curves, Polygons and Symmetry
11.3 Curves, Polygons and Symmetry Polygons Simple Definition A shape is simple if it doesn t cross itself, except maybe at the endpoints. Closed Definition A shape is closed if the endpoints meet. Polygon
More information11 th Annual Harvard-MIT Mathematics Tournament
11 th nnual Harvard-MIT Mathematics Tournament Saturday February 008 Individual Round: Geometry Test 1. [] How many different values can take, where,, are distinct vertices of a cube? nswer: 5. In a unit
More information1 Solution of Homework
Math 3181 Dr. Franz Rothe February 4, 2011 Name: 1 Solution of Homework 10 Problem 1.1 (Common tangents of two circles). How many common tangents do two circles have. Informally draw all different cases,
More information