Question

The perimeter of an equilateral triangle is $6$. The length of an altitude drawn on any of its sides is ..............

• A. $2\sqrt{3}$
• B. $\frac{\sqrt{3}}{2}$
• C. $2$
• D. $\sqrt{3}$

Answer 1

The correct option is D $\sqrt{3}$

Perimeter of $△ABC=6$ .......... $\left(i\right)$
All sides of an equilateral triangle are equal. Let the measure of each side be $x$.
Then $3x=6$ ..... From $\left(i\right)$
$⟹x=2$

Let $AD$ be the altitude drawn on $BC$ from $A$.

Then $D$ is the midpoint of $BC⟹BD=DC=\frac{1}{2}\times BC=1$.
Now, in $△ADC,\angle ADC={90}^o$
$\therefore {AD}^2+{CD}^2={AC}^2$ ....... [By Pythagoras theorem]
$⟹{AD}^2+1^2=2^2$
$⟹{AD}^2=4-1=3$
$\therefore AD=\sqrt{3}$

Hence, the answer is $\sqrt{3}$.