Question

The sides of a triangle are in the ratio 13:14:15 and its perimeter is 84 cm. Find the area of the triangle.
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Answer 1

$\huge\mathbb{SOLUTION:-}$

$\begin{cases}\sf{Sides\:of\: triangle\:in\: ratio}\\ \sf{13:14:15\:\:\:its\: perimeter=84cm}\end{cases}$

• We have to find the area of triangle

• Now let the side of triangle be x

$\boxed{\sf{Perimeter\:of\: triangle=sum\:of\:all\:sides}}$

$\bf\underline{\underline{\red{According\:To\: Question:-}}}$

• The side of triangle be 13 x, 14x and 15x

$\sf\longrightarrow 84=13x+14x+15x\\ \\ \sf\longrightarrow 84=42x\\ \\ \sf\longrightarrow \cancel{\frac{84}{42}}=x\\ \\ \sf\implies x=2$

• The value of x is 2

• Now the sides be
• 13×2=26cm
• 14×2=28cm
• 15×2=30cm

$\sf\underline{Sides\:of\: triangle=26cm,\:28cm\:and\: 30cm}$

• Now we have to find the area of triangle

• The sides of triangle are different so here use heron's formula to find the area of triangle

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

• Where s is the semi perimeter

$\tt\implies s=\frac{a+b+c}{2}$

$\bold{we\:have}\begin{cases}\sf{Sides\:of\: triangle}\\ \sf{26cm\:28cm\:and\:30cm}\end{cases}$

• Let's find the value of semi perimeter

$\sf\implies s=\frac{26+28+30}{2}\\ \\ \sf\implies s=\cancel{\frac{84}{2}}=42\\ \\ \tt\implies{\blue{semi\: perimeter=42cm}}$

$\sf\hookrightarrow Area\:of\:triangle=\sqrt{42(42-26)(42-28)(42-30)}\\ \\ \sf\hookrightarrow Area\:of\:\triangle=\sqrt{42\times 16\times 14\times 12}\\ \\\sf\hookrightarrow Area\:of\:\triangle=\sqrt{7\times 6\times 4\times 4\times 7\times 2\times 6\times 2}\\ \\ \sf\hookrightarrow Area\:of\:\triangle=7\times 4\times 6\times 2\\ \\ \sf\hookrightarrow Area\:of\:\triangle=28cm\times 12cm\\ \\ \sf\implies Area\:of\: triangle=336cm{}^{2}$

$\boxed{\sf{\purple{Area\:of\: triangle=336cm{}^{2}}}}$

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