Sec 3: Coordinate Geometry

University:
Stanford University
Course:
Introduction to Mathematical Thinking
Academic year:

2023
Views:

261

Pages:

4
Author:

MAlphen

Coordinate Geometry 1 The diagram shows a quadrilateral ABCD. The coordinates of A and B are (2,8) and (8,6) respectively. M is the midpoint of AB and CM is perpendicular to AB. The equation of BC is 3𝑦 = 4𝑥 – 14. The point D lies on the y-axis and 𝐷𝐴𝐵 = 90˚. [3] [5] [2] (i) Find the coordinates of D. (ii) Find the coordinates of C. (iii) Find the coordinates of ABCD. 3 Solutions to this question by accurate drawing will not be accepted. In the trapezium OPQR, the point P has coordinates (2,4) and the point R has coordinates (5,0). The sides OP and QR are parallel, and PQ is perpendicular to OP. . (i) Show that the coordinates of Q are (6,2) (ii) Find the equation of the perpendicular bisector of OP and explain if the [5] perpendicular bisector cuts the line segment QR. [3] (iii) T is a point which lies on the perpendicular bisector of OP such that the [3] area of quadrilateral ORPT IS 25 units2. Find the coordinates of T. 1 Solutions: 𝑀 𝐴𝐵 =− 𝑀𝐴𝐷 = 3 8−𝑎 =3 2−0 𝑎=2 𝐷(0,2) (i) Let 𝐷(0,2𝑎) 8 | 1 3 At (5,7), 𝑐 = −8 𝑦 = 3𝑥 − 8 4 14 𝑦= 𝑥− 3 3 4 14 𝑥− = 3𝑥 − 8 3 3 𝑥=2 𝑦 = −2 𝐶(2, −2) (ii) 𝑦 = 5𝑥2+ 𝑐 1 (iii) 0 2 | 2 8 6 −2 2 8 1 = |(12 − 16 + 4) − (64 + 12 + 4)| 2 = 40 units2 Ans: (i) 𝐷(0,2) (ii) 𝐶(2, −2) (iii) 40 units2 3 Solutions: (i) Gradient of 𝑂𝑃 = 4 2 =2 1 Equation of PQ: 4 = − (2) + 𝑐 2 𝑐=5 1 Equation of PQ is 𝑦 = − 𝑥 + 5 ------- (1) 2 Equation of QR: 0 = 2(5) + 𝑐 𝑐 = −10 Equation of QR is 𝑦 = 2𝑥 − 10 -------(2) 1 Sub (1) into (2):− 𝑥 + 5 = 2𝑥 − 10 2 15 𝑥 = 2.5 =6 Sub 𝑥 = 6 into (2): 𝑦 = 12 − 10 =2 Thus, coordinates of Q is (6, 2). (shown) (ii) Midpoint of OP = ( 0+2 0+4 , ) 2 2 = (1,2) Equation of perpendicular bisector of OP: 1 2 = − (1) + 𝑐 𝑐=2 2 1 2 Equation of perpendicular bisector of OP is 1 1 𝑦 = − 𝑥 + 2 --------- (3) 2 2 1 1 2 2 Sub (3) into (2): − 𝑥 + 2 = 2𝑥 − 10 12.5 𝑥= =5 Sub 𝑥 = 5 into (2): 𝑦 = 2(5) − 10 =0 Thus the perpendicular bisector of OP cuts the line segment QR at R(5,0). 2.5 Ans: (i) Sub (1) into (2) = 6, Sub 𝑥 = 6 into (2) = 2, (ii) Sub (3) into (2) = 5, Sub 𝑥 = 5 into (2) = 0, (iii) 𝑥 =– 5, 𝑦 = 5. Coordinates of T is (– 5𝑎, 5)