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Find the area of the shaded region in the figure.

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{cm}^{2}

384

{cm}^{2}

480

{cm}^{2}

None

answer is B.

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Detailed Solution

Area of a triangle is given by

\sqrt{s (s - a) (s - b) (s - c)}

.
Using the Pythagoras’s theorem in right angled triangle ADB, we get

{AB}^{2} = {AD}^{2} + {BD}^{2} = 16^{2} + 12^{2} = 256 + 144 = 400

\Rightarrow AB = 20

cm
Area of triangle ABD =

\frac{1}{2} \times AD \times BD = \frac{1}{2} \times 12 \times 16 = 96 {cm}^{2}

The perimeter of a triangle is the sum of the lengths of all the three sides.
The lengths of the three sides of the triangle ABC are 20 cm, 48 cm, and 52 cm.
Perimeter of triangle ABC=20+48+52=120 cm
s = 60 cm
Area of triangle ABC =

\sqrt{60 (60 - 20) (60 - 48) (60 - 52)} = \sqrt{60 \times 40 \times 12 \times 8} = 12 \times 8 \times 5 = 480 {cm}^{2}

⇒Area of shaded region=Area of triangle ABC - Area of triangle ABD
⇒Area of shaded region=480 - 96 = 384

{cm}^{2}

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