Q16. - Geometry

Question

Find the area of each figure:

An equilateral triangle with height  9

Step-by-Step Solution

Answer

The area of given figure is $27\sqrt{3}$ square units.

Step 1. Given information.

An equilateral triangle with height  9

Here,  $h=9$

Step 2. Find the side of triangle.

We can use Pythagorean Theorem. 

Pythagorean Theorem define that in a right angle triangle, the square of the hypotenuse is equal to the sum of squares of other sides.

$H^2=P^2+B^2$

$a^2=h^2+{\left(\frac{a}{2}\right)}^2$

$a^2-\frac{a^2}{4}=h^2$

$\frac{3a^2}{4}=h^2$

$\frac{3a^2}{4}=h^{2}3a^2=4h^2$

$a=\frac{2}{\sqrt{3}}h$

$a=\frac{2}{\sqrt{3}}\times 9$

$a=6\sqrt{3}$

Step 3. Find the area of given figure.

$A=\frac{\sqrt{3}}{4}{a}^2$

    $=\frac{\sqrt{3}}{4}{\left(6\sqrt{3}\right)}^2$

     $=\frac{108\sqrt{3}}{4}$

  $=27\sqrt{3}$Square units.

Therefore, the area of given figure is$27\sqrt{3}$square units.