MCQ
In the given figure, ABCD is a rectangle inscribed in a circle having length 8 cm and breadth 6 cm. If $\pi=3.14$ then the area of the shaded region is:
- A$264cm^2$
- B$266cm^2$
- C$272cm^2$
- ✓None of the above
✓
Answer
Correct option: D.
None of the above
(D). All options are incorrect; the correct answer is 30.5cm.
Solution:
Join AC.
Now, AC is the diameter of the circle.
We have:
$AC^2= AB^2+ BC^2$ [By pythagoras' theorem]
$\Rightarrow\text{AC}^2=\big\{(8)^2+(6)^2\big\}\text{cm}^2$
$\Rightarrow\text{AC}^2=(64+36)\text{cm}^2$
$\Rightarrow\text{AC}^2=100\text{cm}^2$
$\Rightarrow\text{AC}=10\text{cm}$
$\therefore$ Radius of the circle $=\frac{10}{2}\text{cm}$
$=5\text{cm}$
Now,
Area of the shaded region = Area of the circle with radius 5cm - Area of the rectangle ABCD
$=\big|(3.14\times5\times5)-(8\times6)\big|\text{cm}^2$
$=\Big|\Big(\frac{314}{100}\times25\Big)-48\Big|\text{cm}^2$
$=\Big(\frac{157}{2}-48\Big)\text{cm}^2$
$=\frac{61}{2}\text{cm}^2$
$=30.5\text{cm}^2$
Solution:
Join AC.
Now, AC is the diameter of the circle.
We have:
$AC^2= AB^2+ BC^2$ [By pythagoras' theorem]
$\Rightarrow\text{AC}^2=\big\{(8)^2+(6)^2\big\}\text{cm}^2$
$\Rightarrow\text{AC}^2=(64+36)\text{cm}^2$
$\Rightarrow\text{AC}^2=100\text{cm}^2$
$\Rightarrow\text{AC}=10\text{cm}$
$\therefore$ Radius of the circle $=\frac{10}{2}\text{cm}$
$=5\text{cm}$
Now,
Area of the shaded region = Area of the circle with radius 5cm - Area of the rectangle ABCD
$=\big|(3.14\times5\times5)-(8\times6)\big|\text{cm}^2$
$=\Big|\Big(\frac{314}{100}\times25\Big)-48\Big|\text{cm}^2$
$=\Big(\frac{157}{2}-48\Big)\text{cm}^2$
$=\frac{61}{2}\text{cm}^2$
$=30.5\text{cm}^2$
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