coordinates of a b distance of ab 45slau.weebly.com/uploads/1/3/3/0/13308739/_y9supp_3e09_b.pdffind...

© 2010 Chung Tai Educational Press. All rights reserved.

9.13

1. Match the coordinates of A and B with their corresponding distances.

Coordinates of A and B Distance of AB

A(0, 0), B(3, 2) • • 45 units

A(4, −3), B(1, 3) • • 104 units

A(−5, 2), B(5, 4) • • 34 units

A(−1, −5), B(−4, −10) • • 13 units

2. For each of the following, find the distance between P and Q. (Give your answers correct to 3 significant figures.)

(a) P(0, 0), Q(4, 2) (b) P(5, 4), Q(2, 2)

(c) P(3, 3), Q(7, 7) (d) P(5, 2), Q(3, 8)

3. For each of the following, find the distance between E and F. (Express your answers in surd form.)

(a) E(−4, 10), F(3, −2) (b) E(2, −1), F(7, −5)

(c) E(−8, 4), F(−6, −3) (d) E(−9, −4), F(2, −3)

4. The vertices of ΔABC are A(0, 0), B(1, 1) and C(4, 2). Find the perimeter of ΔABC. (Give your

answer correct to 3 significant figures.)

5. The vertices of ΔABC are A(0, 0), B(3, 4) and C(5, 7). Find the perimeter of ΔABC. (Give your answer correct to 3 significant figures.)


9.14

Q (3, 6)

x

y

P (4, 2)

R (−8, −1)O

O

y

xB (2, 0)

C (3, −4)

A (11, −2)

6. The vertices of quadrilateral ABCD are A(−2, −3), ,)4 ,27( B C(5, −2) and .)5 ,

21( −−D Find the

perimeter of ABCD. (Give your answer correct to 3 significant figures.)

7. C is a point on the x-axis. If the respective distances from A(2, 1) and B(−4, 5) to C are equal, find the coordinates of C.

8. In the figure, the vertices of ΔPQR are P(4, 2), Q(3, 6) and R(−8, −1). Prove that PQ ⊥ PR.

9. In the figure, the vertices of ΔABC are A(11, −2), B(2, 0) and C(3, −4).

(a) Prove that AC ⊥ BC.

(b) Hence find ∠BAC. (Give your answer correct to 3 significant figures.)

10. It is given that A(a, 0), B(0, 3) and C(3, 4) are three vertices of square ABCD. If A lies on the positive x-axis, find the value of a.

11. Four points A(1, −6), B(−7, −5), C(−3, 2) and D(3, 0) are given.

(a) Prove that ABCD is a kite.

(b) Prove that AC ⊥ BD.

(c) Hence find the area of ABCD.


9.15

12. Match the following coordinates of A and B with their corresponding slopes.

Coordinates of A and B Slope of AB

A(4, 2), B(2, 3) • •

3

A(−5, 3), B(−4, 6)

• •

61

A(0, 0), B(3, 5)

• •

35

A(−2, −3), B(4, −2)

• •

21

−

13. For each of the following, find the slope of the straight line which passes through the two given points.

(a)

OA (1, 1)

B (3, 3)

x

y (b)

A (2, 1)

B (4, 4)

y

xO

(c)

A (−1, −2)

B (4, 3)

y

xO

14. For each of the following, find the slope of the straight line which passes through points A and B.

(a) A(3, 4), B(5, 6) (b) A(2, 3), B(−4, 5)

(c) A(−3, 1), B(6, 2) (d) A(−2, −1), B(−4, −3)

(e) A(0, 3), B(−7, 2) (f) A(−5, 2), B(3, 0)

(g) A(−4, −1), B(32 , −2) (h) A(−1, −2), B(

21

− , 32

− )


9.16

O

y

x

A (−3, 4)

B (5, −2)

P (3, k)

15. The following are the coordinates of the vertices of line segments AB, CD, EF and GH. Arrange the line segments in ascending order of their slopes.

AB: A(0, 5), B(5, 0); CD: C(0, 3), D(5, 1); EF: E(1, 9), F(3, 4); GH: G(2, 4), H(3, 1)

16. If the slope of the straight line passing through points A(2, 4) and B(5, a) is 1, find the value of a.

17. If the slope of the straight line passing through points A(a, −3) and B(2a, 3) is

21

− , find the value of a.

18. Prove that each of the following sets of points are collinear.

(a) A(−3, 2), B(0, 2), C(4, 2)

(b) D(−3, 4), E(−3, 0), F(−3, 5)

(c) L(1, −1), M(2, −2), N(−7, 7)

(d) P(0, 3), Q(1, 1), R(3, −3)

19. If P(−2, −10), Q(0, −4) and R(a, 2) are collinear, find the value of a.

20. It is given that the slope of the straight line passing through )34 ,0( A and B(2, −2) is equal to the

slope of the straight line passing through M(−4, 8) and )31 ,( −xN . Find the value of x.

21. Three points A(−3, 4), B(5, −2) and P(3, k) are given. If the slope of AB is equal to the sum of the slopes of AP and BP, find the value of k.


9.17

O

y

x

A (−3, 3)

B (6, −2)

P (3, k)

22. Three points A(−3, 3), B(6, −2) and P(3, k) are given. If the slope of BP is equal to the product of the slopes of AB and AP, find the value of k.

23. It is given that A(a, b) is a point on the graph of the equation y = 3x − 5.

(a) Express b in terms of a.

(b) Hence express the coordinates of A in terms of a.

(c) If A is the point of intersection of the graph of the equation y = 3x − 5 and the straight line passing through P(13, 12) and Q(−7, −4), find the coordinates of A by using the result of (b).

24. Which of the following straight lines are parallel lines?

Straight line Points lying on the straight line

L1 (2, 2), (4, 4)

L2 (1, 0), (2, 1)

L3 (4, 2), (2, 4)

L4 (3, −1), (−3, 1)

L5 (5, 6), (6, 7)

25. L1 is a straight line with the slope of −4. L2 is a straight line passing through points A(−2, 3) and B(−3, 7). Prove that L1 // L2.

26. Four points A(−3, −2), B(−4, −5), C(6, 5) and D(7, 8) are given. Prove that AD // BC.


9.18

27. It is given that L1 and L2 are two parallel lines. If the slope of L2 is −2, and L1 passes through P(4, 9) and Q(1, b), find the value of b.

28. It is given that L1 and L2 are two parallel lines. If the slope of L1 is k, and L2 passes through

A(3, 2k − 11) and B(5, 9), find the value of k.

29. If the straight line passing through points A(4, b) and B(3, 2) is parallel to the straight line passing through points C(−1, 2) and D(3, 5) , find the value of b.

30. If the straight line passing through points A(−2, 3) and B(a, −4) is parallel to the straight line passing through points C(6, −3) and D(−5, −10) , find the value of a.

31. Three vertices of parallelogram ABCD are A(3, 5), B(−3, −2) and C(4, −1). If point D lies in quadrant I, find the coordinates of D.

32. The vertices of quadrilateral PQRS are ,)7 ,49( −−P ,)2 ,

49( −Q R(−5, 2) and S(−5, −7).

(a) Find the slope of each side of quadrilateral PQRS.

(b) Find the length of each side of quadrilateral PQRS.

(c) What kind of quadrilateral is PQRS?

33. It is given that L1 and L2 are the graphs of the equations 3x − 6y = 0 and x − 2y + 8 = 0 respectively.

(a) If (2, a) and (−4, b) are points on L1, find the values of a and b.

(b) If (p, 4) and (q, −6) are points on L2, find the values of p and q.

(c) Hence prove that L1 and L2 are parallel lines.


9.19

34. It is given that straight line L is the graph of the equation 7x + 6y − 5 = 0 and A(a, b) is a point on L.

(a) Express the coordinates of A in terms of a.

(b) Three points B(−2, 4), C(−5, −2) and D(−1, −6) are given. If ABCD is a trapezium where AB // CD, find the coordinates of A by using the result of (a).

35. Which of the following straight lines are perpendicular lines?

Straight line Points lying on the straight line

L1 (1, 1), (−2, −2)

L2 (4, 3), (4, −2)

L3 (3, 4), (1, 6)

L4 (4, −5), (−2, −5)

L5 (2, 4), (3, −3)

36. 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(2, 4), B(3, 6)

(b) P(9, −7), Q(−4, −3)

37. Two points A(7, −1) and B(8, 1) are given, and the slope of CD is

21

− . Prove that CDAB ⊥ .

38. Four points A(5, 0), B(4, 3), C(−2, 6) and D(−5, 5) are given. Prove that AB ⊥ CD.


9.20

O

y

x

L1L2

A (4, 7)

B (−2, 0) C

O

y

x

A (3, 7)

C (10, 0)B

L1L2

39. The vertices of ΔABC are A(4, −2), B(−3, 2) and C(8, 5). Prove that ABC is a right-angled triangle.

40. If the straight line passing through A(5, 2) and B(a, 9) is perpendicular to a straight line with the

slope of 73 , find the value of a.

41. It is given that straight lines L1 and L2 are perpendicular to each other. If the slope of L1 is −3, and L2

passes through points A(a, −5) and B(14, −1), find the value of a.

42. It is given that straight lines L1 and L2 are perpendicular to each other. If the slope of L1 is k, and L2

passes through points P(6, −1) and Q(2k, −6), find the value of k.

43. Two points A(5, 2) and B(−3, 4) are given, and C is a point lying on the y-axis. Find the coordinates of C if AB ⊥ AC.

44. Four points A(3, 8), B(−21, 19), C(a, b) and D(9, −5) are given. Find the coordinates of C if AC ⊥ BD and AD // BC.

45. In the figure, straight lines L1 and L2 intersect perpendicularly at A(4, 7), and L1 and L2 intersect the x-axis at B(−2, 0) and C respectively.

(a) Find the slope of L1. (b) Find the slope of L2. (c) Find the coordinates of C. (d) Hence find the area of ΔABC.

46. In the figure, straight lines L1 and L2 intersect perpendicularly at A(3, 7), and L1 and L2 intersect the x-axis at B and C(10, 0) respectively.

(a) Find the coordinates of B. (b) Hence find the area of ΔABC.


9.21

47. It is given that straight lines L1 and L2 are the graphs of the equations x + 3y + 1 = 0 and 6x − 2y − 8 = 0 respectively.

(a) If (5, a) and (b, −3) are points on L1, find the values of a and b. (b) If (p, 2) and (9, q) are points on L2, find the values of p and q. (c) Hence prove that L1 and L2 are perpendicular lines.

48. It is given that A(a, b), B(9, 4) and C(6, −2) satisfy the following conditions:

Condition I: Slope of AB112

=

Condition II: AC ⊥ BC

(a) According to condition I, express a in terms of b. (b) According to condition II, express 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 2x − 3y + 10 = 0.

(a) Express the coordinates of A in terms of a.

(b) If A(a, b), B(2, 12), C(−6, 2) and D(−1, −2) form a rectangle, find the coordinates of A by using the result of (a).

50. In each of the following figures, P is the mid-point of line segment AB. Find the coordinates of P.

(a)

A (1, 2)

O

B (5, 8)

P

y

x

(b) y

xA (0, 0)

O

B (7, 4)

P

(c)

B (10, 1)O

A ( , 5)

P

12

y

x


9.22

51. For each of the following, M is the mid-point of line segment AB. Find the coordinates of M.

(a) A(−2, 4), B(6, 8)

(b) A(5, −3), B(3, 6)

(c) A(−4, −2), B(−6, 0)

(d) )2 ,4

13( ),5 ,41( BA −

52. In each of the following figures, P is a point on line segment AB. Find the coordinates of P.

(a)

AP : PB = 2 : 1O

P

A (1, 1)

B (10, 5)

x

y (b)

AP : PB = 2 : 3

OP

A (−5, −6)

B (4, 2)

x

y (c)

AP : PB = 1 : 4

O

P

A (−2, 4)

B (4, −4)

x

y

53. For each of the following, P is a point on line segment AB. Find the coordinates of P.

(a) A(0, 4), B(3, −5); AP : PB = 1 : 2

(b) 2:1: ;)4 ,29( ,)6 ,

25( =−− PBAPBA

54. Given that P is the mid-point of A(−6, 4) and B(k, 8), and the x-coordinate of P is 4, find the value of k.

55. If P(4, 2) is the mid-point of A(−2, a) and B(b, 8), find the values of a and b.

56. If P(1, 1) is the mid-point of A(a + 7, 2a) and B(4 − 2a, b), find the values of a and b.

57. If P(−3, −7) is the mid-point of A(2a, a − 5) and B(5b + 1, 4b − 1), find the values of a and b.


9.23

A (14, 16)

D (−12, 6)

E

OF

B (19, 3)

C (− 7, −7)

y

x

x

y

O

D

B (−2, −8)

A (3, 2)

C (5, −4)

E

PR : RQ = 1 : r

P (−2, 1)

R

O B (3, 0)A (−1, 0)

Q (4, 7)

x

y

58. Two points A(7, 3) and B(−7, 10) are given. If C(k + 1, 8) divides line segment AB into two parts in the ratio of 1 : r, find the values of r and k.

59. In the figure, ABCD is a rectangle. E is the mid-point of AD. F divides BC into two parts in the ratio of 3 : 2.

(a) Find the coordinates of E and F.

(b) Find the perimeter of quadrilateral CDEF. (Give your answer correct to 1 decimal place.)

60. In the figure, E divides line segments AB and CD into two parts in the ratio of 2 : 3 respectively.

(a) Find the coordinates of E.

(b) Find the coordinates of D.

(c) Are AC and DB parallel to each other?

61. In the figure, R divides PQ into two parts in the ratio of 1 : r. The area of ΔABR is 5 square units.

(a) If AB is the base of ΔABR, express the height of ΔABR in terms of r.

(b) Hence find the value of r.


9.24

A (8, 11)

B (2, −1)

C (20, 5)D (4, a)

E (b, c)

PA : AQ = 1 : r

x

y

O

Q (7, 5)

P (−3, 1)

A B

CD

O

P (−a, a)R (b, b)

Q (b − a, b + a)

x

y

O

C (k, k)

x

y

B (h + k, k)

A (h, 0)

62. In the figure, C and D are points on the x-axis. A divides line segment PQ into two parts in the ratio of 1 : r. ABCD is a square with the area of 9 square units.

(a) Express the length of each side of square ABCD in terms of r.

(b) Hence find the value of r.

63. In the figure, D and E are points on AB and AC respectively, and DE // BC.

(a) Find AD : DB. (b) Find the coordinates of D and E. (c) Find the ratio of the areas of ΔABC and ΔADE.

64. Prove that OPQR in the following figure is a rectangle.

65. Prove that OABC in the following figure is a parallelogram.


9.25

OD

B (b, 0)

A (a, 2a)

C (2a − b, 0)x

y

O

B

A (a, 0)

C (c, c + 1)

x

y

D

O

G (5k, k)

D (0, 2k)EB

A (0, 4k)

x

y

y

xO

A

C (c, 0) B (b, 0)

66. The figure shows ΔOAD. B and C are points on OD such that OB = CD.

(a) Find the coordinates of D.

(b) Hence prove that OA = DA.

(c) What type of triangle is OAD?

67. The figure shows quadrilateral OABC. AC and BO intersect at D. AB = OC and OA // CB.

(a) Find the coordinates of B.

(b) Find the x-coordinate of D.

(c) Hence, if the y-coordinate of D is half of its x-coordinate, prove that OAD is an isosceles triangle.

68. The figure shows ΔAOG. B and D are points on AO, and E is a point on AG such that DE // OG and BE // DG. It is given that the x-coordinate and y-coordinate of E are the same.

(a) Find the coordinates of B and E.

(b) Prove that AB × DO = AD × BD.

69. In the figure, OAB and OAC are right-angled triangles, where C is a point on OB.

(a) Find the coordinates of A.

(b) Hence prove that (i) AC2 = OC × CB. (ii) AB2 = OB × BC. (iii) OA2 = OB × OC.


9.26

70. In the figure, OD and BC are the medians of ΔOAB and intersect at E. OE : ED = r : 1 and BE : EC = m : 1.

OE : ED = r : 1�BE : EC = m : 1

O

A (2a, 2b)

E

B (6a, 0)

DC

x

y

(a) Find the coordinates of C and D.

(b) Prove that the coordinates of E are )1

,14(

rrb

rra

++.

(c) Prove that the coordinates of E are )1

,1

)6((

mmb

mam

++

+ .

(d) Hence find the values of r and m.

coordinates of a b distance of ab 45slau.weebly.com/uploads/1/3/3/0/13308739/_y9supp_3e09_b.pdffind...

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