#### Question

In the given figure,

\[\square\] PQRS is a rectangle. If PQ = 14 cm, QR = 21 cm, find the areas of the parts x, y, and z.

#### Solution

PQRS is a rectangle.

∴ ∠Q = ∠R = 90º

Radius of sector PTQ = PQ = 14 cm

∴ Area of part x = Area of the sector PTQ = \[\frac{\theta}{360° } \times \pi r^2 = \frac{\angle Q}{360° } \times \pi \times \left( PQ \right)^2 = \frac{90° }{360° } \times \frac{22}{7} \times \left( 14 \right)^2\] = 154 cm^{2}

Radius of sector TUR = TR = QR − QT = QR − PQ = 21 − 14 = 7 cm (QT = PQ)

∴ Area of part y = Area of the sector TUR = \[\frac{\theta}{360° } \times \pi r^2 = \frac{\angle R}{360° } \times \pi \times \left( TR \right)^2 = \frac{90°}{360° } \times \frac{22}{7} \times \left( 7 \right)^2\] = 38.5 cm^{2 }

Now,

Area of rectangle PQRS = QR × PQ = 21 × 14 = 294 cm^{2}

∴ Area of part z = Area of rectangle PQRS − Area of part x − Area of part y = 294 − 154 − 38.5 = 101.5 cm^{2}