# (b) the equation of the perpendicular bisector of AB. [3]

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1 HORIZON EDUCATION SINGAPORE Additional Mathematics Practice Questions: Coordinate Geometr 1 Set 1 1 In the figure, ABCD is a rhombus with coordinates A(2, 9) and C(8, 1). The diagonals AC and BD cut at E. (i) Calculate the co-ordinates of E. [1] (ii) Find the equation of BD. [2] It is given that the equation of AD is = 0. (iii) Find the equation of BC. [2] (iv) Calculate the length of AB. [3] 2 The line 3 = 7 intersects the curve = 7 at A and B. Find (a) the coordinates of the points A and B, [3] (b) the equation of the perpendicular bisector of AB. [3] 3 The line = 2 intersects the curve = + 1 at points A and B. Find the equation of the perpendicular bisector of the line AB. [6] 4 Two points A and B have coordinates (3 2, 2 5) and ( 2 5, 2) respectivel. Without the use of a calculator, calculate the gradient of the line AB, leaving our answer in the form a + b 10, where a and b are integers. [4] Page 1 of 12

2 5 Solution to this question b accurate drawing will not be accepted. B (6, 13) C A D (1, ) The diagram shows a trapezium ABCD in which AB is parallel to DC. The point A lies on the -ais. Points B and D are (6, 13) and (1, 2) respectivel. AB C = BC D = 90. Given that the equation of DC is 3 = 4 10, find (a) the coordinates of A, [3] (b) the coordinates of C, [4] (c) the area of the trapezium ABCD. [3] 6 Solutions to this question b accurate drawing will not be accepted. The coordinates of P and Q are ( 1, 10) and (11, 6) respectivel. (i) Find the equation of the perpendicular bisector of PQ. [3] (ii) Given that there is a pair of coordinates of point A which meets the perpendicular bisector of PQ at the -ais, find coordinates of A. [3] (iii) If PAQB is a parallelogram, find the coordinates of point B. [3] Page 2 of 12

3 7 Solutions to this question b accurate drawing will not be accepted. A(7, 10) D. M(8, 6) B C The diagram shows a rhombus ABCD in which A is (7, 10) and D is on the -ais. The point M(8, 6) is the midpoint of AC. Find (a) the coordinates of C, [2] (b) the coordinates of D, [3] (c) the coordinates of X given that X is on MD such that area of triangle AXC = 1 area of triangle ADC. [3] 4 8 Solutions to this question b accurate drawing will not be accepted. C (2, 1) B 0 D A ( 1, 4) In the quadrilateral ABCD, the points A and C are ( 1, 4) and (2, 1) respectivel. The line BC is parallel to 4 1 = 0 and perpendicular to AB. The foot of the perpendicular from A to CD bisects CD and the rest on the -ais with the -coordinate 3. Find (i) the coordinates of B and of D, (ii) the radius of the circle which passes through A, B and C. [8] Page 3 of 12

4 9 Solution to this question b accurate drawing will not be accepted. PQRS is a trapezium in which PQ is parallel to SR and PS is perpendicular to both PQ and SR. The coordinate of P, Q and R are (0, 11), (3, 2) and (13, 12) respectivel. S P(0, 11) R (13, 12) Q(3, 2) Find (a) the equation of SR, [2] (b) the coordinates of the point S, [2] T is a point on RS produced such that PQRT is a parallelogram. Find (c) the coordinates of T, [2] (d) the ratio RS: ST, [2] (e) the shortest distance of R from PQ. [2] 10 ABCD is a rectangle, where A is ( 3, 0) and C is (1, 7). Given that the equation of AB is 3 = 2 + 6, find (i) the equation of BC, [2] (ii) the coordinates of B, [2] (iii) the coordinates of D, [2] (iv) the area of ABCD. [2] Page 4 of 12

5 11 The diagram, which is not drawn to scale, shows a right-angled triangle PQR in which QPR = 90 and the coordinates of P and Q are (3, 5) and ( 1, 3) respectivel. Given that the gradient of QR is 1 and the the perpendicular from P 2 to QR to meets QR at S, find P (3, 5) S R O Q( 1, 3) (a) the equation of QR and of PR, [3] (b) the coordinates of R and of S, [5] (c) the ratio, QS: SR, [1] (d) the numerical value of area of ΔPSR area of ΔPQR. [1] Page 5 of 12

6 12 In the quadrilateral ABCD, the points A, B and D are (3, 3), (0, 1) and (6, 2) respectivel. The line BD bisects the line AC at right angles at the point M. Find the coordinates of M and of C. [8] A (3, 3) D (6, 2) B (0, 1) M C 13 The straight line + 2 = 5 intersects the curve = 29 at the points A and B. Given that A lies below the -ais, and that P lies on AB such that the area of ΔAOP is 1 of the area of ΔAOB, where O is the origin, find the 4 coordinates of P. [6] Page 6 of 12

7 14 Solutions to this question b accurate drawing will not be accepted. The diagram shows a kite OABC whose diagonals meet at M. The coordinates of A, B and C are (11, a), (5, 3) and (c, c + 3) respectivel, where a and c are constants. C(c, c + 3) M B(5, 3) O A(11, a) Find (a) the coordinates of M, [1] (b) the equation of AC, [3] (c) the value of a, [1] (d) the coordinates of C, [2] (e) the area of the kite OABC. [3] 15 P, Q, R and S are the points (0, 9), ( 3, 3), (1, 2) and (3, 7) respectivel. (i) Find the area of the quadrilateral PQRS. [2] (ii) Find the equation of the line, l 1, that is perpendicular to PS and passes through the point R. [3] (iii) If PQ is etended to meet the line l 1 at T, find the ratio of PQ: QT. [4] 16 A point M lies on the line 2 + = 10 and is at a distance of 5 units from the origin (0, 0). Find the possible -coordinates of M. [3] Page 7 of 12

8 17 ABCD is a rectangle such that A, B and C are the points (0, 1), (t, 2t + 1) and (4, 4) respectivel. (i) Show that the value of t is 2. [2] (ii) Find the equation of CD. [2] (iii) Find the coordinates of the point of intersection of the 2 diagonals. [2] (iv) Find the equation of the perpendicular bisector of AB. [3] 18 Solutions to this question b accurate drawing will not be accepted. The line 4 3 = 1 intersects the curve = at the points P and Q. (a) Find the coordinates of P and of Q. [4] (b) Find the equation of the perpendicular bisector of PQ. [3] It is given that the perpendicular bisector of PQ intersects the -ais at the point R. (c) Find the distance of R from PQ. [3] 19 In the given diagram, point A is located on the -ais and the equation of the straight line AB is 2 = + 8. B (p, q) A O (a) Write the coordinates of the midpoint of AB in terms of p and q. [2] (b) Show that the equation of the perpendicular bisector of AB is = 2p + q + 4. [2] (c) If the perpendicular bisector of AB intersects the -ais at = 14, find the value of p and of q. [3] 20 The line 5 + = 9 intersects the curve = 5 at the points P and Q. Find the midpoint of PQ. [6] Page 8 of 12

9 21 Solution to the question b accurate drawing will not be accepted. A (2, 9) B = 2 O The diagram shows a right-angled triangle ABO in which O is the origin, A is the point (2, 9) and AB O = 90. The equation of the line OB is = 2. Find (i) the equation of the line AB, [2] (ii) the coordinates of B. [2] C is a point on the perpendicular bisector of OA and is such that BC is parallel to the -ais. (iii) Find the coordinates of C. [3] area of ΔOAB D lies on AB produced such that = 1. area of ΔOAD 3 (iv) Find the coordinates of D. [3] Page 9 of 12

10 22 Solutions to this question b accurate drawing will not be accepted. The diagram shows a trapezium ABCD in which the coordinates of A and C are (2, 1) and (3, 4) respectivel Given that E is a point on the -ais, such that ABCE is a square. D C (3, 4) E O A(2, 1) B (i) Find the coordinates of E and B. [4] (ii) Find the equation of AB. [2] (iii) Given that the area of square ABCE is 4 times the area of triangle CDE, find the coordinates of D. [3] Page 10 of 12

11 23 Solutions to this question b accurate drawing will not be accepted. A (6, 9) C (8,4) E (p, ) 0 B (6, 1) D The diagram shows an isosceles triangle ABC in which A is the point (6, 9). B is the point (6, 1). It is given that the area of the triangle ABC is 8 units 2. (i) Find the coordinates of C. [2] The line CB is etended to the point D such that the area of triangle ADC is thrice the area of triangle ABC. (ii) Find the coordinates of D. [3] A line is drawn from A, parallel to CD, to the point E (p, ). (iii) Find the value of p. [2] (iv) Determine whether AED = 90. [3] Page 11 of 12

12 24 Solutions to this question b accurate drawing will not be accepted. The diagram shows a quadrilateral ABCD in which A is (3, 0), C is (3a, 5a + 3) and D is ( 2, 5). The equation of AB is 5 = 3 9 and angle ADC = 90. C (3a, 5a + 3) D ( 2, 5) B A (3, 0) (i) Find the value of a. If F is the foot of the perpendicular bisector of CD from B, find (ii) (iii) the coordinates of F, the coordinates of B, and (iv) the area of the quadrilateral ABCD. [11] 25 The line 2 + = 5 intersects the cruve 2 + = 6 at the points A and B. Find the equation of the perpendicular bisector of AB. [6] 26 The points A and B have coordinates ( 8, 10) and ( 2, 2). Find the equation of the perpendicular bisector of AB. [4] END Page 12 of 12

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