Question

triangle abc is an equilateral triangle with side lengths labeled a, b, and c. which expressions represent the area of triangle abc? choose three correct answers.
$\frac{absin(60^{circ})}{2}$
$\frac{a^{2}bsin(60^{circ})}{2}$
$acsin(60^{circ})$

Explanation:

Step1: Recall area - formula for a triangle

The area formula for a triangle is $A=\frac{1}{2}ab\sin C=\frac{1}{2}bc\sin A=\frac{1}{2}ac\sin B$, where $a$, $b$, $c$ are the side - lengths of the triangle and $A$, $B$, $C$ are the opposite angles respectively.

Step2: Identify angles and sides for equilateral triangle

In an equilateral triangle $ABC$, $a = b = c$ and $A=B = C=60^{\circ}$.

Step3: Substitute into the area formula

Substituting into $A=\frac{1}{2}ab\sin C$, when $C = 60^{\circ}$, we get $A=\frac{ab\sin(60^{\circ})}{2}$. Also, since $b = c$, substituting into $A=\frac{1}{2}ac\sin B$ with $B = 60^{\circ}$ gives $A=\frac{ac\sin(60^{\circ})}{2}$. And since $a = b = c$, we can also write the area as $\frac{bc\sin(60^{\circ})}{2}$, $\frac{a^{2}\sin(60^{\circ})}{2}$, $\frac{b^{2}\sin(60^{\circ})}{2}$, $\frac{c^{2}\sin(60^{\circ})}{2}$.

Answer:

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