Question

PQR is a right angled triangle where angle PRQ is a right angle. The length of PQ is (2x+ 7) m, the length of PR is (3x + 3) m and the length of QR is (x + 2) m

(a) Use the diagram to form an equation involving x, and show that it reduces to x ^2 - x - 6 = 0

Answer 1

Explanation:

Using Pythagoras' Theorem, hypotenuse squared is equal to adjacent squared plus opposite squared.

The hypotenuse here is line PQ, the adjacent is line PR and the opposite is line QR.

Therefore, PQ^2 = PR^2 + QR^2

(2x + 7)^2 = (3x + 3)^2 + (x + 2)^2

{4x}^{2} + 28x + 49 = {9x}^{2} + 18x + 9 + {x}^{2} + 4x + 4

collecting like terms

{4x}^{2} - {9x}^{2} - {x}^{2} + 28x - 18x - 4x + 49 - 9 - 4 = 0

- {6x}^{2} + 6x + 36 = 0

Multiplying through by -1 to match the given equation

{6x}^{2} - 6x - 36 = 0

dividing through by 6

{x}^{2} - x - 6 = 0 (the final equation)