Question

find area of shaded region in figure, where AD=12CM, BD=16CM, AC=52CM, BC=48CM

Answer 1

Area of shaded region = area of ∆ABC - area of ∆ADB.
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$\textbf{Find area of triange ADB:}$

∆ADB is a right angle triangle.

In ∆ADB,
AD = 12 cm
BD = 16 cm

Ar(∆ADB) = 1/2 × base × height

=> Ar(∆ADB) = 1/2 × AD × BD

=> Ar(∆ADB) = 1/2 × 12 × 16

=> Ar(∆ADB) = 96 cm^2
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$\textbf{Find area of triange ABC}$

you can find AB by using Pythagoras theorem.
${AB}^{2} = {AD}^{2} +{ BD}^{2} \\ \\ {AB}^{2} = {12}^{2} + {16}^{2} \\ \\ {AB}^{2} = 144 + 256 = 400 \\ \\ AB = \sqrt{400} \\ \\ AB = 20 \: cm$

The sides of the triangle are 48cm, 52cm and 20cm.

$s = \frac{48 + 52 + 20}{2} = \frac{120}{2} = 60 \: cm$

$\text{Area of triangle ABC} = \sqrt{s(s - a)(s - b)(s - c)} \\ \\ = \sqrt{60 \times (60 - 52)(60 - 48)(60 - 20) }$

$= \sqrt{60 \times 8 \times 12 \times 40} \\ \\ = \sqrt{230400} \\ \\ = 480 \: {cm}^{2}$
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Area of shaded region = area of ∆ABC - area of ∆ADB.

=> Area of shaded region = 480 - 96

=> Area of shaded region = 384 cm^2