a. In the diagram above P, Q, R, and S are points on the circle with centre O.
QR // OS , ∠QOR = 2m, ∠QPR = n and ∠SOR = 54°. Find the values of m and n.
b. The length of a rectangle is 4 cm more than the width. If the perimeter is 40 cm, find the area.
Explanation
a. 2m = 2 × n (the angle at the centre is twice the angle at the circumference)
∴ m = n
∠SOR = ∠QRO = 54° (alternate angles are equal)
From ∆ROQ
Since |OQ| = |OR| (radii), therefore, ∆ROQ is an isosceles triangle
2m = 180° - 2(54°) = 180° - 108° (sum of angles in a ∆ is 180°)
2m = 72°
m = \(\frac{72}{2}\) = 36°
Recall: m = n
∴ m = n = 36°
b. Let the length of the rectangle be L and the width be W
L = W + 4 (given)
Perimeter = 2(L + W)
40 = 2(L + W)
40 = 2((W + 4) + W) (Recall: L = W + 4)
40 = 2(2W + 4)
40 = 4W + 8
40 - 8 = 4W
32 = 4W
W = \(\frac{32}{4}\)
W = 8cm
L = W + 4 = 8 + 4
L = 12cm
Area of a rectangle = L x W
= 12 × 8
= \(96 cm^2\)