Cambridge International AS and A Level Mathematics Pure Mathematics 1

Co-ordinate geometry

P1^

2

Method 2

Pythagoras’ theorem states that for a right-angled triangle whose hypotenuse has

length a and whose other sides have lengths b and c, a^2 = b^2 + c^2.

Conversely, if you can show that a^2 = b^2 + c^2 for a triangle with sides of lengths a, b,

and c, then the triangle has a right angle and the side of length a is the hypotenuse.

This is the basis for the alternative proof, in which you use

length^2 = (x 2 − x 1 )^2 + (y 2 − y 1 )^2.

PQ^2 = (3 − 2)^2 + (2 − 7)^2 = 1 + 25 = 26

RP^2 = (2 − 0)^2 + (7 − 5)^2 = 4 + 4 = 8

RQ^2 = (3 − 0)^2 + (2 − 5)^2 = 9 + 9 = 18

Since 26 = 8 + 18, PQ^2 = RP^2 + RQ^2

⇒ Sides RP and RQ are at right angles.

EXERCISE 2A 1 For the following pairs of points A and B, calculate:

(a) the gradient of the line AB

(b) the mid-point of the line joining A to B

(c) the distance AB

(d) the gradient of the line perpendicular to AB.

(i) A(0, 1) B(2, −3) (ii) A(3, 2) B(4, −1)

(iii) A(−6, 3) B(6, 3) (iv) A(5, 2) B(2, −8)

(v) A(4, 3) B(2, 0) (vi) A(1, 4) B(1, −2)

2 The line joining the point P(3, −4) to Q(q, 0) has a gradient of 2.

Find the value of q.

3 The three points X(2, −1), Y(8, y) and Z(11, 2) are collinear (i.e. they lie on the

same straight line).

Find the value of y.

4 The points A, B, C and D have co-ordinates (1, 2), (7, 5), (9, 8) and (3, 5).

(i) Find the gradients of the lines AB, BC, CD and DA.

(ii) What do these gradients tell you about the quadrilateral ABCD?

(iii) Draw a diagram to check your answer to part (ii).

5 The points A, B and C have co-ordinates (2, 1), (b, 3) and (5, 5), where b> 3

and ∠ABC = 90°. Find:

(i) the value of b

(ii) the lengths of AB and BC

(iii) the area of triangle ABC.