Find the 2. 2 perimeter of ABCD. (Give your answer correct to 3 significant figures.)
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1 7 6. The vertices of quadrilateral ABC are A(, ), B (, 4), C(5, ) and (, 5). Find the perimeter of ABC. (Give our answer correct to significant figures.). Match the coordinates of A and B with their corresponding distances. Coordinates of A and B istance of AB 7. C is a point on the -ais. If the respective distances from A(, ) and B( 4, 5) to C are equal, find the coordinates of C. A(0, 0), B(, ) 45 units A(4, ), B(, ) 04 units 8. In the figure, the vertices of ΔQR are (4, ), Q(, 6) and R( 8, ). rove that Q R. Q (, 6) A( 5, ), B(5, 4) 4 units A(, 5), B( 4, 0) units R ( 8, ) (4, ). For each of the following, find the distance between and Q. (Give our answers correct to significant figures.) (a) (0, 0), Q(4, ) (b) (5, 4), Q(, ) (c) (, ), Q(7, 7) (d) (5, ), Q(, 8). For each of the following, find the distance between and F. (press our answers in surd form.) (a) ( 4, 0), F(, ) (b) (, ), F(7, 5) (c) ( 8, 4), F( 6, ) (d) ( 9, 4), F(, ) 9. In the figure, the vertices of ΔABC are A(, ), B(, 0) and C(, 4). (a) rove that AC BC. (b) Hence find BAC. (Give our answer correct to significant figures.) B (, 0) C (, 4) A (, ) 0. It is given that A(a, 0), B(0, ) and C(, 4) are three vertices of square ABC. If A lies on the positive -ais, find the value of a. 4. The vertices of ΔABC are A(0, 0), B(, ) and C(4, ). Find the perimeter of ΔABC. (Give our answer correct to significant figures.) 5. The vertices of ΔABC are A(0, 0), B(, 4) and C(5, 7). Find the perimeter of ΔABC. (Give our answer correct to significant figures.). Four points A(, 6), B( 7, 5), C(, ) and (, 0) are given. (a) rove that ABC is a kite. (b) rove that AC B. (c) Hence find the area of ABC. 00 Chung Tai ducational ress. All rights reserved Chung Tai ducational ress. All rights reserved.
2 . Match the following coordinates of A and B with their corresponding slopes. Coordinates of A and B Slope of AB A(4, ), B(, ) A( 5, ), B( 4, 6) 6 A(0, 0), B(, 5) 5 A(, ), B(4, ). For each of the following, find the slope of the straight line which passes through the two given points. (a) (b) (c) 5. The following are the coordinates of the vertices of line segments AB, C, F and GH. Arrange the line segments in ascending order of their slopes. AB: A(0, 5), B(5, 0); C: C(0, ), (5, ); F: (, 9), F(, 4); GH: G(, 4), H(, ) 6. If the slope of the straight line passing through points A(, 4) and B(5, a) is, find the value of a. 7. If the slope of the straight line passing through points A(a, ) and B(a, ) is, find the value of a. 8. rove that each of the following sets of points are collinear. (a) A(, ), B(0, ), C(4, ) (b) (, 4), (, 0), F(, 5) (c) L(, ), M(, ), N( 7, 7) (d) (0, ), Q(, ), R(, ) 9. If (, 0), Q(0, 4) and R(a, ) are collinear, find the value of a. A (, ) B (, ) A (, ) B (4, 4) A (, ) B (4, ) 4 0. It is given that the slope of the straight line passing through A ( 0, ) and B(, ) is equal to the slope of the straight line passing through M( 4, 8) and N (, ). Find the value of. 4. For each of the following, find the slope of the straight line which passes through points A and B. (a) A(, 4), B(5, 6) (b) A(, ), B( 4, 5) (c) A(, ), B(6, ) (d) A(, ), B( 4, ) (e) A(0, ), B( 7, ) (f) A( 5, ), B(, 0) (g) A( 4, ), B(, ) (h) A(, ), B(, ). Three points A(, 4), B(5, ) and (, k) are given. If the slope of AB is equal to the sum of the slopes of A and B, find the value of k. A (, 4) B (5, ) (, k) 00 Chung Tai ducational ress. All rights reserved Chung Tai ducational ress. All rights reserved.
3 . Three points A(, ), B(6, ) and (, k) are given. If the slope of B is equal to the product of the slopes of AB and A, find the value of k. A (, ) B (6, ) 7. It is given that L and L are two parallel lines. If the slope of L is, and L passes through (4, 9) and Q(, b), find the value of b. (, k). It is given that A(a, b) is a point on the graph of the equation = 5. (a) press b in terms of a. (b) Hence epress the coordinates of A in terms of a. (c) If A is the point of intersection of the graph of the equation = 5 and the straight line passing through (, ) and Q( 7, 4), find the coordinates of A b using the result of (b). 8. It is given that L and L are two parallel lines. If the slope of L is k, and L passes through A(, k ) and B(5, 9), find the value of k. 9. If the straight line passing through points A(4, b) and B(, ) is parallel to the straight line passing through points C(, ) and (, 5), find the value of b. 0. If the straight line passing through points A(, ) and B(a, 4) is parallel to the straight line passing through points C(6, ) and ( 5, 0), find the value of a.. Three vertices of parallelogram ABC are A(, 5), B(, ) and C(4, ). If point lies in quadrant I, find the coordinates of. 4. Which of the following straight lines are parallel lines? Straight line oints ling on the straight line L (, ), (4, 4) L (, 0), (, ) L (4, ), (, 4) L 4 (, ), (, ) L 5 (5, 6), (6, 7) 9 9. The vertices of quadrilateral QRS are (, 7), Q (, ), R( 5, ) and S( 5, 7). 4 4 (a) Find the slope of each side of quadrilateral QRS. (b) Find the length of each side of quadrilateral QRS. (c) What kind of quadrilateral is QRS? 5. L is a straight line with the slope of 4. L is a straight line passing through points A(, ) and B(, 7). rove that L // L. 6. Four points A(, ), B( 4, 5), C(6, 5) and (7, 8) are given. rove that A // BC.. It is given that L and L are the graphs of the equations 6 = 0 and + 8 = 0 respectivel. (a) If (, a) and ( 4, b) are points on L, find the values of a and b. (b) If (p, 4) and (q, 6) are points on L, find the values of p and q. (c) Hence prove that L and L are parallel lines. 00 Chung Tai ducational ress. All rights reserved Chung Tai ducational ress. All rights reserved.
4 4. It is given that straight line L is the graph of the equation = 0 and A(a, b) is a point on L. (a) press the coordinates of A in terms of a. (b) Three points B(, 4), C( 5, ) and (, 6) are given. If ABC is a trapezium where AB // C, find the coordinates of A b using the result of (a). 9. The vertices of ΔABC are A(4, ), B(, ) and C(8, 5). rove that ABC is a right-angled triangle. 40. If the straight line passing through A(5, ) and B(a, 9) is perpendicular to a straight line with the slope of, find the value of a It is given that straight lines L and L are perpendicular to each other. If the slope of L is, and L passes through points A(a, 5) and B(4, ), find the value of a. 5. Which of the following straight lines are perpendicular lines? Straight line oints ling on the straight line L (, ), (, ) L (4, ), (4, ) L (, 4), (, 6) L 4 (4, 5), (, 5) 4. It is given that straight lines L and L are perpendicular to each other. If the slope of L is k, and L passes through points (6, ) and Q(k, 6), find the value of k. 4. Two points A(5, ) and B(, 4) are given, and C is a point ling on the -ais. Find the coordinates of C if AB AC. 44. Four points A(, 8), B(, 9), C(a, b) and (9, 5) are given. Find the coordinates of C if AC B and A // BC. L 5 (, 4), (, ) 6. For each of the following, find the slope of the straight line which is perpendicular to the line passing through the two given points. (a) A(, 4), B(, 6) (b) (9, 7), Q( 4, ) 45. In the figure, straight lines L and L intersect perpendicularl at A(4, 7), and L and L intersect the -ais at B(, 0) and C respectivel. (a) Find the slope of L. (b) Find the slope of L. (c) Find the coordinates of C. (d) Hence find the area of ΔABC. L B (, 0) L A (4, 7) C 7. Two points A(7, ) and B(8, ) are given, and the slope of C is. rove that AB C. 8. Four points A(5, 0), B(4, ), C(, 6) and ( 5, 5) are given. rove that AB C. 46. In the figure, straight lines L and L intersect perpendicularl at A(, 7), and L and L intersect the -ais at B and C(0, 0) respectivel. (a) Find the coordinates of B. (b) Hence find the area of ΔABC. L B L A (, 7) C (0, 0) 00 Chung Tai ducational ress. All rights reserved Chung Tai ducational ress. All rights reserved.
5 5. For each of the following, M is the mid-point of line segment AB. Find the coordinates of M. (a) A(, 4), B(6, 8) 47. It is given that straight lines L and L are the graphs of the equations + + = 0 and 6 8 = 0 respectivel. (a) If (5, a) and (b, ) are points on L, find the values of a and b. (b) If (p, ) and (9, q) are points on L, find the values of p and q. (c) Hence prove that L and L are perpendicular lines. (b) A(5, ), B(, 6) (c) A( 4, ), B( 6, 0) (d) A(, 5), B(, ) It is given that A(a, b), B(9, 4) and C(6, ) satisf the following conditions: Condition I: Slope of AB = Condition II: AC BC (a) According to condition I, epress a in terms of b. (b) According to condition II, epress a in terms of b. (c) Hence find the coordinates of A. 49. It is given that A(a, b) is a point on the graph of the equation + 0 = 0. (a) press the coordinates of A in terms of a. (b) If A(a, b), B(, ), C( 6, ) and (, ) form a rectangle, find the coordinates of A b using the result of (a). 5. In each of the following figures, is a point on line segment AB. Find the coordinates of. (a) (b) (c) B (4, ) A (, 4) B (0, 5) A (, ) A : B = : A ( 5, 6) B (4, 4) A : B = : A : B = : 4 5. For each of the following, is a point on line segment AB. Find the coordinates of. (a) A(0, 4), B(, 5); A : B = : 5 9 (b) A (, 6), B(, 4); A : B = : 50. In each of the following figures, is the mid-point of line segment AB. Find the coordinates of. (a) (b) B (5, 8) A (, ) (c) B (7, 4) A (0, 0) A (, 5) B (0, ) 54. Given that is the mid-point of A( 6, 4) and B(k, 8), and the -coordinate of is 4, find the value of k. 55. If (4, ) is the mid-point of A(, a) and B(b, 8), find the values of a and b. 56. If (, ) is the mid-point of A(a + 7, a) and B(4 a, b), find the values of a and b. 57. If (, 7) is the mid-point of A(a, a 5) and B(5b +, 4b ), find the values of a and b. 00 Chung Tai ducational ress. All rights reserved Chung Tai ducational ress. All rights reserved.
6 58. Two points A(7, ) and B( 7, 0) are given. If C(k +, 8) divides line segment AB into two parts in the ratio of : r, find the values of r and k. 59. In the figure, ABC is a rectangle. is the mid-point of A. F divides BC into two parts in the ratio of :. (a) Find the coordinates of and F. (b) Find the perimeter of quadrilateral CF. (Give our answer correct to decimal place.) (, 6) C ( 7, 7) F A (4, 6) B (9, ) 6. In the figure, C and are points on the -ais. A divides line segment Q into two parts in the ratio of : r. ABC is a square with the area of 9 square units. (a) press the length of each side of square ABC in terms of r. (b) Hence find the value of r. 6. In the figure, and are points on AB and AC respectivel, and // BC. (a) Find A : B. (b) Find the coordinates of and. (c) Find the ratio of the areas of ΔABC and ΔA. A B (, ) C A : AQ = : r A (8, ) (4, a) B (, ) Q (7, 5) (b, c) C (0, 5) 60. In the figure, divides line segments AB and C into two parts in the ratio of : respectivel. (a) Find the coordinates of. (b) Find the coordinates of. A (, ) (c) Are AC and B parallel to each other? C (5, 4) B (, 8) 64. rove that QR in the following figure is a rectangle. Q (b a, b + a) ( a, a) R (b, b) 6. In the figure, R divides Q into two parts in the ratio of : r. The area of ΔABR is 5 square units. (a) If AB is the base of ΔABR, epress the height of ΔABR in terms of r. R Q (4, 7) 65. rove that ABC in the following figure is a parallelogram. (b) Hence find the value of r. (, ) A (, 0) B (, 0) R : RQ = : r C (k, k) A (h, 0) B (h + k, k) Chung Tai ducational ress. All rights reserved. 00 Chung Tai ducational ress. All rights reserved.
7 70. In the figure, and BC are the medians of ΔAB and intersect at. : = r : and B : C = m :. 66. The figure shows ΔA. B and C are points on such that B = C. A (a, a) A (a, b) (a) Find the coordinates of. C (b) Hence prove that A = A. (c) What tpe of triangle is A? B (6a, 0) B (b, 0) C (a b, 0) : = r : B : C = m : (a) Find the coordinates of C and. 67. The figure shows quadrilateral ABC. AC and B intersect at. AB = C and A // CB. (a) Find the coordinates of B. (b) Find the -coordinate of. (c) Hence, if the -coordinate of is half of its -coordinate, prove that A is an isosceles triangle. C (c, c + ) B A (a, 0) 4ra rb (b) rove that the coordinates of are (, ). + r + r (6 + m) a mb (c) rove that the coordinates of are (, ). + m + m (d) Hence find the values of r and m. 68. The figure shows ΔAG. B and are points on A, and is a point on AG such that // G and B // G. It is given that the -coordinate and -coordinate of are the same. (a) Find the coordinates of B and. (b) rove that AB = A B. A (0, 4k) B (0, k) G (5k, k) 69. In the figure, AB and AC are right-angled triangles, where C is a point on B. A (a) Find the coordinates of A. (b) Hence prove that (i) AC = C CB. (ii) AB = B BC. (iii) A = B C. C (c, 0) B (b, 0) 00 Chung Tai ducational ress. All rights reserved Chung Tai ducational ress. All rights reserved.
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