Question

If the height of an equilateral triangle is $9$, what is its area?

• A. $27$
• B. $54$
• C. $81$
• D. $27\sqrt{3}$
• E. $54\sqrt{3}$

Answer 1

The correct option is D $27\sqrt{3}$

Knowing that the triangle is equilateral makes everything easier: the heights are also medians.
Let the sides of triangle be a and h be the median
the triangle can be divided into two congruent right triangles and according to pythagorams theoram $a^2=h^2+{\left(\frac{a}{2}\right)}^2$ ,so $\left(\frac{3}{4}\right)a^2=h^2$
i.e $a^2=\left(\frac{4}{3}\right)h^2$
thus we get that side is given by $a=\frac{2\sqrt{3}h}{3}=\frac{2\sqrt{3}}{3}\times 9=6\sqrt{3}$d
now the area
$A=\frac{a\times h}{2}$$=\frac{\frac{2\sqrt{3}h}{3}h}{2}$$=\frac{\sqrt{3}}{3}{h}^2=$$\frac{\sqrt{3}}{3}81=27\sqrt{3}$