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

The correct option is D $9\sqrt{3}$

Since the triangle is an equilateral triangle, its angles are 60. In addition, the vertical line AX bisects the angle BAC, so it splits it into two congruent angles of ${30}^o$ each. Therefore, the two smaller triangles are 30-60-90 triangles. Thus, the ratio of the lengths of the sides is$1$:$\sqrt{3}$ :$2$. Use this to fill in the information of the figure as follows.
The area of a triangle is given by the formula $A=\frac{1}{2}bh$, where b is the length of the base and h is the height. In the figure, the base is 3 + 3 = 6 and the height is $3\sqrt{3}$. Substitute these values into the formula and calculate the area.
$A=\frac{1}{2}\left(6\right)\left(3\sqrt{3}\right)=9\sqrt{3}$