Question : In the given figure, ABCD is a square. EFGH is a square formed by joining mid-points of sides of ABCD. LMNO is a square formed by joining mid-points of sides of EFGH. A circle is inscribed inside LMNO. If the area of a circle is 38.5 cm2 then what is the area (in cm2) of square ABCD?
Option 1: 98
Option 2: 196
Option 3: 122.5
Option 4: 171.5
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Correct Answer: 196
Solution :
The area of a circle,
$ \text{Area} = \pi r^2$ where $r$ is the radius of the circle.
Given that the area of the circle is 38.5 cm
2
.
$⇒ r = \sqrt{\frac{\text{Area}}{\pi}} = \sqrt{\frac{38.5}{\pi}} = 3.5$ cm
The side length of square LMNO = $2r = 7$ cm
In $\triangle NHO$,
$ON^2=NH^2+HO^2$
$⇒7^2=NH^2+HO^2$
$⇒49=2NH^2$ $(\because NH=HO)$
$⇒NH=HO=\frac{7}{\sqrt2}$ cm
The side length of square $FEGH=2NH = 7\sqrt2$ cm
In $\triangle FDH$,
$HF^2=FD^2+DH^2$
$⇒(7\sqrt2)^2=FD^2+DH^2$
$⇒98=2FD^2$ $(\because FD=DH)$
$⇒ FD=DH = 7$ cm
The side length of square $ABCD=2 FD=14$ cm
So, the area of the square $ABCD=196$ cm
2
Hence, the correct answer is 196.
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