Question

The figure below is an equilateral triangle with sides of length 6. What is the area of the triangle?

A. 12
B. 18
C. 36
D. $9\sqrt 3 $

Answer 1

Hint:
In an equilateral triangle, the perpendicular divides the opposite side in two equal parts. Use Pythagoras theorem to find the height of the given triangle. Then, use the formula, $\dfrac{1}{2} \times {\text{base}} \times {\text{height}}$ to find the required area.

Complete step by step solution:
Label the points on the given triangle.

ABC is an equilateral triangle and AD is perpendicular on BC.
In an equilateral triangle, the perpendicular divides the opposite side in two equal parts.
Hence, the value of $DC$ is 3cm.
We will find the height of the triangle using Pythagoras theorem.
$A{C^2} = A{D^2} + D{C^2}$
On substituting the values, we will get,
$
   \Rightarrow {6^2} = A{D^2} + {3^2} \\
   \Rightarrow 36 = A{D^2} + 9 \\
   \Rightarrow A{D^2} = 27 \\
   \Rightarrow AD = 3\sqrt 3 \\
$
The area of the triangle is $\dfrac{1}{2} \times {\text{base}} \times {\text{height}}$
Therefore, the area of the required triangle is \[\dfrac{1}{2} \times 6 \times 3\sqrt 3 = 9\sqrt 3 \]

Hence, option D is correct.

Note:
We can also calculate the area of an equilateral triangle using the formula, $\dfrac{{\sqrt 3 }}{4}{a^2}$, where $a$ is the length of each side of an equilateral triangle.
$\dfrac{{\sqrt 3 }}{4}{\left( 6 \right)^2} = 9\sqrt 3 $ square units.