Question

Find the perimeter of an equilateral triangle whose area is equal to that of triangle with side 21 CM 16 cm and 13 cm answer correct to 2 decimal place.​

Answer 1

$\bf{\Huge{\underline{\underline{\rm{ANSWER\::}}}}}}$

12√15 cm

$\bf{Given}\begin{cases}\sf{1st\:Side\:of\:\triangle\:=\:21cm}\\ \sf{2nd\:side\:of\:\triangle\:=\:16cm}\\ \sf{3rd\:side\:of\:\triangle\:=\:13cm}\end{cases}}$

$\bf{\large{\underline{\underline{\sf{To\:find\::}}}}}$

The perimeter of an equilateral triangle.

$\bf{\Large{\underline{\underline{\tt{\blue{Explanation\::}}}}}}$

Using formula of the Heron's Formula :

$\bullet\sf{Semi-Perimeter\:of\:\triangle(S)\:=\:\frac{A+B+C}{2} }\\\\\bullet\sf{Area\:of\:\triangle\:=\:\sqrt{S(S-A)(S-B)(S-C)} }$

A/q

$\implies\sf{Semi-perimeter\:=\:\dfrac{21cm+16cm+13cm}{2} }\\\\\\\\\implies\sf{Semi-perimeter\:=\:\cancel{\dfrac{50cm}{2} }}\\\\\\\\\implies\sf{\blue{Semi-perimeter\:\big[S\big]=\:25cm}}\\\\\\\tt{Now,}\\\\\implies\sf{Area\:of\:\triangle\:=\:\sqrt{S(S-A)(S-B)(S-C)} }\\\\\\\\\implies\sf{Area\:of\:\triangle\:=\:\sqrt{25(25-21)(25-16)(25-13)} }\\\\\\\\\implies\sf{Area\:of\:\triangle\:=\:\sqrt{25(4)(9)(12)} }\\\\\\\\\implies\sf{Area\:of\:\triangle\:=\:\sqrt{25*4*9*12}}$

$\implies\sf{Area\:of\:\triangle\:=\:\sqrt{5*5*2*2*3*3*2*2*3} }\\\\\\\\\implies\sf{Area\:of\:\triangle\:=\:(5*2*3*2\sqrt{3} )cm^{2}}$

$\implies\sf{\blue{Area\:of\:\triangle\:=\:60\sqrt{3}\: cm^{2} }}$

According to the question :

$\bf{\Large{\boxed{\sf{Area\:of\:an\:equilateral\:\triangle\:=\:\frac{\sqrt{3} }{4} a^{2} }}}}}}$

$\implies\sf{\dfrac{\sqrt{3} }{4} a^{2} \:=\:60\sqrt{3} }\\\\\\\\\implies\sf{a^{2} \:=\:\dfrac{60\cancel{\sqrt{3} }*4}{\cancel{\sqrt{3}} } }\\\\\\\\\implies\sf{a^{2} \:=\:(60*4)cm^{2} }\\\\\\\\\implies\sf{a\:=\:\sqrt{240cm^{2} } }\\\\\\\\\implies\sf{\blue{a\:=\:4\sqrt{15} \:cm}}$

Now,

$\bf{\large{\boxed{\sf{Perimeter\:of\:an\:equilateral\:\triangle\:=\:Side+Side+Side}}}}}}$

$\longmapsto\sf{Perimeter\:=\:4\sqrt{15}\:cm +4\sqrt{15} \:cm+4\sqrt{15} \:cm}\\\\\\\\\longmapsto\sf{\blue{Perimeter\:=\:12\sqrt{15}\: cm}}$

Thus,

Perimeter is 12√15 cm.

Answer 2

Step-by-step explanation:

Sides of a triangle:

a = 21cm

b = 16cm

c = 13cm

Area of the triangle = √s(s-a)(s-b)(s-c)

Where s = a+b+c / 2

s = 21+16+13 / 2

s = 50/2 = 25

Area = √25(25-21)(25-16)(25-13) = 103.923 cm²

Area of the equilateral triangle = √3/4 a²

√3/4 a² = 103.923

a = 15.59 cm

So, Perimeter of the equilateral triangle = 3×side = 3×15.59 = 46.77 cm

Therefore, the perimeter of equilateral triangle = 46.77cm