Question

the triangle shown is an equilateral triangle.
what is the area of the equilateral triangle with the length of each side equal to ( a )?
( \frac{1}{2}asin(60^circ) )
( a^2sin(60^circ) )
( 3asin(60^circ) )
( \frac{1}{2}a^2sin(60^circ) )

Explanation:

Step1: Recall triangle area formula

For any triangle with two sides $b,c$ and included angle $\theta$, area is $\frac{1}{2}bc\sin\theta$.

Step2: Substitute values for equilateral triangle

In this case, $b = c = a$, $\theta = 60^\circ$.
Expression: $\text{Area} = \frac{1}{2} \times a \times a \times \sin(60^\circ) = \frac{1}{2}a^2\sin(60^\circ)$

Answer:

$\frac{1}{2}a^2\sin(60^\circ)$