Question

The height of an equilateral triangle measures 9√3.find it's area

Answer 1

$\Longrightarrow{\boxed{\bold{Answer:81\sqrt{3}\:unit^2 }}}$

$\textbf{\underline{Step-by-step explanation :- }}$

➣ Height of equilateral triangle = √3a/2

➣ Height of equilateral triangle = 9√3 units

9√3 units = √3a/2

9√3 × 2/√3 = a

18 units = a

➣ Now, area of equilateral triangle = √3a²/4

➣ Here, a = 18

= √3/4 × 18²

= √3/4 × 18 × 18

$\bold{\huge{=\underline{81\sqrt{3}\:unit^2}}}$

Answer 2

Refer the attachment for figure.

Before we proceed :-

The height, median and angle bisector, all are same for an equilateral triangle. Hence, the height will divide the third side in two equal parts.

Now all sides are equal in Ann equilateral triangle. Let the side be 'a'. So the bisected side would be a/2


Now apply Pythagoras Theorem. By Pythagoras Theorem,

$( \frac{a}{2} ) {}^{2} + ( 9\sqrt{3} ) {}^{2} = a {}^{2}$


=>>
$\frac{ {a}^{2} }{4} + 243 = {a}^{2} \\ \\ = > 243 = {a}^{2} - \frac{ {a}^{2} }{4} \\ \\ = > 243 = \frac{ {4a}^{2} }{4} - \frac{ {a}^{2} }{4} \\ \\ = > 243 = \frac{3 {a}^{2} }{4} \\ \\ = > {a}^{2} = 243 \times \frac{4}{3} \\ \\ = > {a}^{2} = 324 \\ \\ = > a = 18$

So the side of the triangle is 18 unit.

So the area = 1/2 × base × height

= 1/2 × 18 × 9√3

= 9 × 9√3

= 81√3 square units

Answer :- 81√3