Question

Find the length of the side and perimeter of an equilateral triangle whose height is $\sqrt{3}$ cm.

Answer 1

Since, ABC is an equilateral triangle, CD is the perpendicular bisector of AB.

Now, According to Pythagoras theorem,
In ∆ACD

$$\begin{gathered} {\mathrm{AC}}^2={\mathrm{AD}}^2+{\mathrm{CD}}^2 \\ \Rightarrow {\left(2a\right)}^2=a^2+{\left(\sqrt{3}\right)}^2 \\ \Rightarrow 4a^2-a^2=3 \\ \Rightarrow 3a^2=3 \\ \Rightarrow a^2=1 \\ \Rightarrow a=1\mathrm{cm} \end{gathered}$$

Hence, the length of the side of an equilateral triangle is 2 cm.

Now,
Perimeter of the triangle = (2 + 2 + 2) cm
= 6 cm

Hence, perimeter of an equilateral triangle is 6 cm.