In the below figure, $ O $ is the centre of a circular arc and $ A O B $ is a straight line. Find the perimeter and the area of the shaded region correct to one decimal place. (Take $ \pi=3.142) $"

Given:

\( O \) is the centre of a circular arc and \( A O B \) is a straight line.

To do: 

We have to find the perimeter and the area of the shaded region correct to one decimal place.

Solution:

A semicircle is drawn on the diameter $AB$
$\triangle ACB$ is drawn in this semicircle.

In right angled triangle $ACB$, by Pythagoras theorem,

$AB=\sqrt{\mathrm{AC}^{2}+\mathrm{BC}^{2}}$

$=\sqrt{(12)^{2}+(16)^{2}}$

$=\sqrt{144+256}$

$=\sqrt{400}$

$=20 \mathrm{~cm}$

Therefore,

Radius of the semicircle $=\frac{20}{2}$

$=10 \mathrm{~cm}$

Perimeter of the shaded region $=$ Perimeter of the semicircle  $+\mathrm{AC}+\mathrm{BC}$

$=\pi r+12+16$

$=3.142 \times 10+28$

$=31.42+28 \mathrm{~cm}$

$=59.42$

$=59.4 \mathrm{~cm}$

Area of the shaded region $=$ Area of the semicircle $-$ Area of $\Delta \mathrm{ABC}$

$=\frac{1}{2} \pi r^{2}-\frac{1}{2} \times 12 \times 16$

$=\frac{1}{2} \times 3.142 \times 10^2$

$=\frac{314.2}{2}-96$

$=157.1-96$

$=61.1 \mathrm{~cm}^{2}$

The perimeter and the area of the shaded region correct to one decimal place are $59.4\ cm$ and $61.1\ cm^2$ respectively.

Tutorialspoint

Updated on: 10-Oct-2022

174 Views

Kickstart Your Career

Get certified by completing the course

Get Started