Question

The height of an equilateral triangle is $\sqrt{6}$ cm. Its area is
(a) $2\sqrt{2}{\mathrm{cm}}^2$
(b) $6\sqrt{2}{\mathrm{cm}}^2$
(c) $2\sqrt{3}{\mathrm{cm}}^2$
(d) $3\sqrt{3}{\mathrm{cm}}^2$

Answer 1

(c) $2\sqrt{3}{\mathrm{cm}}^2$

Height of an equilateral triangle$=\frac{\sqrt{3}}{2}a$ (where a is the side of the equilateral triangle)
$$\begin{gathered} \Rightarrow \frac{\sqrt{3}}{2}a=\sqrt{6} \\ \Rightarrow a=\frac{\sqrt{6}}{\sqrt{3}}\times 2 \\ =2\sqrt{\frac{6}{3}} \\ =2\sqrt{2}\mathrm{cm} \end{gathered}$$

∴ Area of the triangle$=\frac{\sqrt{3}}{4}{a}^2=\frac{\sqrt{3}}{4}\times {\left(2\sqrt{2}\right)}^2=2\sqrt{3}{\mathrm{cm}}^2$