Question

Find the area of the triangle with sides 21 cm, 16 cm and 13 cm. Also, find the perimeter of an equilateral
triangle equal in area to this triangle.​

Answer 1

Area of an equilateral triangle = Area of triangle with sides 21cm, 16cm and 13cm.

Now,

Let's find the Area of triangle with given sides.

We know that,

$\sf{ s = \dfrac{a + b + c}{2}}$

$\sf{ = \dfrac{21 + 16 + 13}{2}}$

$\sf{ = \dfrac{50}{2}}$

$\sf{ = 25}$

Using Heron's Formula,

$\sf{Area = \sqrt{s(s - a)(s - b)(s - c)}}$

$\sf {= \sqrt{25(25 - 21)(25 - 16)(25 - 13)}}$

$\sf{ = \sqrt{25 \times 4 \times 9 \times 12}}$

$\sf{ = \sqrt[60]{3}}$

Thus, Area = $\sf{ \sqrt[60]{3}}$

Now,

$\sf{Area \: of \: equilateral \: triangle = \dfrac{ \sqrt{3} }{4} \times a^{2}}$

$\pink\implies$

$\sf{\dfrac{ \sqrt{3} }{4} {a}^{2} = 60 \sqrt{3}}$

Cancelling $\sf\sqrt{3}$ on both sides,

$\sf{ \dfrac{ {a}^{2} }{4} = 60}$

$\sf{ a^{2} = 60 \times 4}$

$\sf{a = \sqrt{240}}$

a = $\sf{ \sqrt[4]{15}}$

Thus, Side of the equilateral triangle is $\sf{ \sqrt[4]{15}}$.

Now,finding the perimeter.

$\sf{Perimeter \: of \: triangle = 3a}$

$\sf{ = 3 \times \sqrt[4]{15}}$

$\sf {= \sqrt[12]{15}}$

Thus,

Perimeter of the equilateral triangle is $\large\sf{ \sqrt[12]{15}}$

$\large\sf\pink{\sqrt[12]{15} }$

Answer 2

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