Question

Assertion (A)
The area of the quadrant of a circle having a circumference of 22 cm is $9\frac{5}{8}{\mathrm{cm}}^2.$

Reason (R)
The area of a sector of a circle of radius r with central angle x° is $\left(\frac{x\times \mathrm{\pi }{r}^2}{360}\right).$

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) Reason (R) true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Answer 1

(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Assertion (A):
Let r be the radius of the circle.
Now,
Circumference of the circle $=2\mathrm{\pi }r$
We have:
$$\begin{gathered} 2\mathrm{\pi }r=22 \\ \Rightarrow 2\times \frac{22}{7}\times r=22 \\ \Rightarrow r=\left(22\times \frac{7}{44}\right)\mathrm{cm} \\ \Rightarrow r=\frac{7}{2}\mathrm{cm} \end{gathered}$$
Area of the quadrant$=\frac{90°}{360°}\mathrm{\pi }{r}^2$
$$\begin{gathered} =\left(\frac{1}{4}\times \frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}\right){\mathrm{cm}}^2 \\ =\frac{77}{8}{\mathrm{cm}}^2 \\ =9\frac{5}{8}{\mathrm{cm}}^2 \end{gathered}$$
Hence, assertion (A) is true.

Reason (R):
The given statement is true.

Assertion (A) is true and reason (R) is the correct explanation of assertion (A).