Question

The height of a triangle is 4 sqrt 3 . What is the perimeter of the equatorial triangle?

Answer 1

The perimeter of the equatorial triangle is 24 units

Solution:

Given that,

An equilateral triangle has an height equal to $4 \sqrt{3}$

The triangle is shown below

From Triangle ABC in the shown figure AD $=4 \sqrt{3}$

Let the sides of the equilateral triangle be ‘a’

AB = BC = a

Since, it is an equilateral triangle we get,

BD = DC = a ÷ 2

Now, using Pythagoras Theorem in Triangle ABD,

The Pythagorean theorem is this: In a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.

$\mathrm{AB}^{2}=\mathrm{BD}^{2}+\mathrm{AD}^{2}$

$\begin{array}{l}{a^{2}=\left(\frac{a}{2}\right)^{2}+(4 \sqrt{3})^{2}} \\\\ {a^{2}-\left(\frac{a}{2}\right)^{2}=(4 \sqrt{3})^{2}}\end{array}$

$\frac{4 a^{2}-a^{2}}{4}=16 \times 3$

$\begin{array}{l}{\frac{3 a^{2}}{4}=16 \times 3} \\\\ {3 a^{2}=192} \\\\ {a^{2}=192 \div 3=64}\end{array}$

a = 8

Hence, the three sides of the triangle are 8 units each

In equilateral traingle, length of all three sides of triangle are equal

So, Perimeter = 3 $\times$ (Length of each side of triangle)

Perimeter = 3 $\times$ 8 = 24

Thus the perimeter of the equatorial triangle is 24 units