Question

Find the area of the region excluding ∆ ABD by herons formula..please help​

Answer 1

Answer:

Area of shaded region = area of ∆ABC - area of ∆ADB.

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Find area of triange ADB:

∆ADB is a right angle triangle.

In ∆ADB,

AD = 12 cm

BD = 16 cm

$\sf \: Ar(∆ADB) = \frac{1}{2} × base × height$

$\sf=> Ar(∆ADB) = \frac{1}{2} × AD × BD$

$\sf=> Ar(∆ADB) = \frac{1}{2} × 12 × 16$

$\sf=> Ar(∆ADB) = 96 cm^2$

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Find area of triange ABC

you can find AB by using Pythagoras theorem.

$\begin{gathered} {AB}^{2} = {AD}^{2} +{ BD}^{2} \\ \\ {AB}^{2} = {12}^{2} + {16}^{2} \\ \\ {AB}^{2} = 144 + 256 = 400 \\ \\ AB = \sqrt{400} \\ \\ AB = 20 \: cm\end{gathered}$

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

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

$\sf{Area \: of \: triangle \: ABC} = \sqrt{s(s - a)(s - b)(s - c)}$

$\sf = \sqrt{60 \times (60 - 52)(60 - 48)(60 - 20) }$

$\begin{gathered}= \sqrt{60 \times 8 \times 12 \times 40} \\ \\ = \sqrt{230400} \\ \\ = 480 \: {cm}^{2}\end{gathered}$

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Area of shaded region = area of ∆ABC - area of ∆ADB.

$\sf \: => Area \: of \: shaded \: region = 480 - 96$

$\sf=> Area \: of \: shaded \: region = 384 cm^2$

Hence This is Answer