Question

1. area of the major sector (with a circle diagram showing radius 15 cm, central angle 135° for the minor sector, and points p, s, q, r)

Explanation:

Step1: Find the central angle of major sector

The total angle around a point is \(360^\circ\). The minor sector has an angle of \(135^\circ\), so the central angle of the major sector (\(\theta\)) is \(360^\circ - 135^\circ=225^\circ\).

Step2: Recall the formula for the area of a sector

The formula for the area of a sector with radius \(r\) and central angle \(\theta\) (in degrees) is \(A = \frac{\theta}{360^\circ}\times\pi r^{2}\). Here, \(r = 15\space\text{cm}\) and \(\theta = 225^\circ\).

Step3: Substitute the values into the formula

Substitute \(r = 15\) and \(\theta=225^\circ\) into the formula:
\[

$$\begin{align*} A&=\frac{225^\circ}{360^\circ}\times\pi\times(15)^{2}\\ &=\frac{5}{8}\times\pi\times225\\ &=\frac{1125\pi}{8}\\ &\approx\frac{1125\times3.14}{8}\\ &\approx\frac{3532.5}{8}\\ &\approx441.5625\space\text{cm}^2 \end{align*}$$

\]

Answer:

The area of the major sector is approximately \(441.56\space\text{cm}^2\) (or \(\frac{1125\pi}{8}\space\text{cm}^2\) in exact form).