Let A B be a line segment of length 2 . Construct a semicircle S … - Vidyadip

MCQ

Let $A B$ be a line segment of length $2$ . Construct a semicircle $S$ with $A B$ as diameter. Let $C$ be the mid-point of the $\operatorname{arc} A B$. Construct another semicircle $T$ external to the $\triangle A B C$ with chord $A C$ as diameter. The area of the region inside the semi-circle $T$ but outside $S$ is

A
$\frac{\pi}{2}$
✓
$\frac{1}{2}$
C
$\frac{\pi}{\sqrt{2}}$
D
$\frac{1}{\sqrt{2}}$

✓

Answer

Correct option: B.

$\frac{1}{2}$

b
(b)

Given,

$A B$ is diameter of circle $S$ and $C$ is the mid-point of arc length of $A B$.

$A C$ is diameter of circle $T$

$A B=2$

$O A=O B=O C=1$

Area of shaded region

$=$ Area of semi-circle $T+$ Area of

$\triangle O A C-$ Area of quadrant of circle $S$

$=\frac{\pi}{2}\left(\frac{\sqrt{2}}{2}\right)^2+\frac{1}{2} \times 1 \times 1-\frac{\pi}{4} \times(1)^2$

$=\frac{\pi}{4}+\frac{1}{2}-\frac{\pi}{4}=\frac{1}{2}$

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