Question

The height of an equalateral triangle measures 9root 3 cm.find it's area ?

Answer 1

Let the side be x

by Pythagoras theorem,

x²=(x/2)²+(9√3)²

x²=x²/4 +(81*3)

x²-x²/4 ={243}


(4x²-x²)/4 =243

3x²=243*4

x²=972/3

x²=324

x=√324

x=18cm

Now,


Area =a²(√3/4)

Area =18*18*(√3/4)

Area =324(√3/4)

Area of equilateral triangle =81√3 cm²



I hope this will help you



-by ABHAY

Answer 2

Consider an equilateral triangle ABC having altitude AD from vertex A to side BC.
∴ AD = 9√3 cm.
Let AB=BC=AC=x
We know that in an equilateral triangle altitude from any vertex will bisect the opposite side.
∴ BD + CD = BC
∴ 2BD = BC
∴2BD = x ................(∵ length of all sides is x.)
∴ BD = x/2.

Now, In Δ ABD
By Pythagoras Theorem, we have

AB² = AD²+BD²
∴ x² = (9√3)² + (x/2)²
∴ x² = 243 + x²/4
∴ x² - x²/4 = 243
∴ 3x² / 4 = 243
∴ x² = 243 * 4 / 3
∴ x² = 324
∴ x = √324
∴ x = +- 18
However the length of the side cannot be negative so we will take the length of the side of the triangle as +18 cm.

Now , we know the formula for the area of the equilateral triangle
$Area = \frac{ \sqrt{3} }{4}* side^{2} \\ = \frac{ \sqrt{3} }{4} * x^{2}$

Area = √3/4 * (18)² = √3 / 4 * 324 = √3 * 81 
= 81√3 cm²
≈ 140.13 cm²

∴ The area of the equilateral triangle is approximately 140.13 cm².

# Dhruvsh
Hope this helps you !!