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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