Question

In the given figure find the area of the shaded region if AC = 24 CM AB = 7cm and O is the centre

Answer 1

The area of shaded region is $\boxed{161.54{\text{ c}}{{\text{m}}^2}}.$

Further explanation:

The formula for area of semi-circle can be expressed as follows,

$\boxed{{\text{Area of semi - circle}} = \frac{1}{2} \times \pi {r^2}}$

Here, $r$ is the radius of the circle.

Given:

The length of side AC is $24{\text{ cm}}.$

The length of side AB is $7{\text{ cm}}.$

Explanation:

In $\Delta {\text{ABC}}$ use the Pythagoras theorem to obtain the length of side BC.

$\begin{aligned}{\text{B}}{{\text{C}}^{\text{2}}} &= {\text{A}}{{\text{C}}^{\text{2}}} + {\text{A}}{{\text{B}}^{\text{2}}}\\{\text{B}}{{\text{C}}^{\text{2}}} &= {7^2} + {24^2}\\{\text{B}}{{\text{C}}^{\text{2}}} &= 49 + 576\\{\text{B}}{{\text{C}}^{\text{2}}}&= 625\\{\text{BC}}&= 25\\\end{aligned}$

The diameter of the circle is $25{\text{ cm}}.$

The radius of the circle is $\dfrac{{25}}{2}{\text{ cm}}.$

The area of the semi-circle can be obtained as follows,

$\begin{aligned}{\text{Area of semicircle}}&= \frac{1}{2} \times \frac{{22}}{7} \times {\left( {\frac{{25}}{2}} \right)^2}\\&= \frac{{13750}}{{56}}\\&= 245.54{\text{ c}}{{\text{m}}^2}\\\end{aligned}$

The area of triangle can be obtained as follows,

$\begin{aligned}{\text{Area of triangle}}&= \frac{1}{2} \times 7 \times 24\\&=12 \times 7\\&= 84{\text{ c}}{{\text{m}}^2}\\\end{aligned}$

Area of shaded region can be obtained as follows,

$\begin{aligned}{\text{Area of shaded region}} &= {\text{Area of semicircle}} - {\text{Area of triangle}}\\&= 245.54 - 84\\&= 161.54{\text{ c}}{{\text{m}}^2}\\\end{aligned}$

The area of shaded region is $\boxed{161.54{\text{ c}}{{\text{m}}^2}}.$

Learn more:

1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Circles

Keywords: area of shaded region, center, AB=7cm, AC=24 cm, figure, radius of circle, arc length, radian, central angle, intercepted, circle, circumference, sector of a circle, minor sector, major sector, segment, angle.

Answer 2

See the attached figure to better understand the problem

we know that

The inscribed angle in a circle measures half of the arc it comprises.

in this problem
the inscribed angle= ∠ACB
and the arc it comprises measures 180°
then
the ∠ACB=180°/2-------> ∠ACB=90°


applying the Pythagorean theorem
AC²+CB²=AB²-------> AB²=24²+7²-------> AB²=625------> AB=25 cm

the diameter of circle is AB
radius=25/2--------> r=12.5 cm

[the area of a half circle]=pi*r²/2------> pi*12.5²/2--------> 245.44 cm²

[area of triangle ABC]=AC*CB/2--------> 24*7/2-------> 84 cm²

[the area of the shaded region]=[the area of a half circle]-[area of triangle ABC]

[the area of the shaded region]=245.44-84-------> 161.44 cm²

the answer is 
the area of the shaded region is 161.44 cm²