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# Find the altitude of an equilateral triangle of side $ 8 \mathrm{~cm} $.

**Given: **

An equilateral triangle of side $8\ cm$.

**To do:**

We have to find the length of altitude of an equilateral triangle.

**Solution:**

$\vartriangle ABC$ is an equilateral triangle. [Given]

$AD \perp BC$ [Given]

In $\vartriangle ADB$ and $\vartriangle ADC$,

$AB=BC$ [Sides of an equilateral triangle are equal]

$\angle 1=\angle 2=90^o$

$AD=AD$ [Common]

Therefore, by RHS congruence criterion,

$\vartriangle ADB\cong\vartriangle ADC$

$\Rightarrow BD=DC$ [CPCT]

This implies,

$BD=DC=\frac{BC}{2}$

$=\frac{8}{2}$

$=4\ cm$

Therefore, by Pythagoras theorem,

In right angled triangle $ADB$,

$AD^2+BD^2=AB^2$

$\Rightarrow AD^2+(4)^2=(8)^2$

$\Rightarrow AD^2=64−16$

$\Rightarrow AD^2=48$

$\Rightarrow AD=\sqrt{48}$

$\Rightarrow AD=4\sqrt{3}\ cm$

Hence, the altitude of the equilateral triangle is $4\sqrt{3}\ cm$.

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