Question

If A is the area of an equilateral triangle of height h, then

(a) A =$\sqrt{3}$ h² (b) $\sqrt{3}$A = h (c) $\sqrt{3}$A = h² (d) 3A = h²

Answer 1

Let a and h be the side and height of the equilateral triangle respectively. Then
$$\begin{gathered} \frac{a^2\sqrt{3}}{4}=\frac{1}{2}\times a\times h \\ \Rightarrow \frac{a\sqrt{3}}{4}=\frac{1}{2}\times h \\ \Rightarrow a=\frac{2h}{\sqrt{3}} \end{gathered}$$
Therefore
$$\begin{gathered} \mathrm{Area}\mathrm{of}\mathrm{equilateral}\mathrm{triangle}\left(A\right)=\frac{a^2\sqrt{3}}{4} \\ \Rightarrow A=\frac{{\left({\displaystyle \frac{2h}{\sqrt{3}}}\right)}^2\sqrt{3}}{4}=\frac{h^2}{\sqrt{3}} \\ \Rightarrow \sqrt{3}A=h^2 \end{gathered}$$
Hence, the correct option is (c).