Question

An equilateral triangle $△ABC$ is given whose perpendicular is $6\sqrt{3}$ cm. what is the perimeter of triangle?

• A. $12$ cm
• B. $36$ cm
• C. $18\sqrt{3}$ cm
• D. $24$ cm

Answer 1

The correct option is B $36$ cm

On applying hypotenuse leg congruency, we get that perpendicular in an equilateral or isosceles triangle divides the triangle into two congruent triangles hence it divides the base into two equal parts.
Here $△ABD\cong △ACD$
Hence, $BD=CD$In the above figure,
applying pythagoras theorem in $△ABD$
$A{B}^2=B{D}^2+A{D}^2$
$a^2=(\frac{a}{2}{)}^2+(6\sqrt{3}{)}^2$
$a^2=\frac{a^2}{4}+36\times 3$
$4a^2-a^2=4\times 36\times 3$
$3a^2=4\times 36\times 3$
$a^2=4\times 36$
$a=2\times 6$
$a=12$ cm

Now, perimeter of equilateral triangle=$3\times \text{side of triangle}$
Perimeter=$3\times 12$
Perimeter=$36$ cm
So, option $\left(b\right)$ correct.