Question : ' O ' is a point in the interior of an equilateral triangle. The perpendicular distance from ' O ' to the sides are $\sqrt{3}~cm, 2 \sqrt{3}~cm, 5 \sqrt{3}~cm$. The perimeter of the triangle is:
Option 1: 48 cm
Option 2: 32 cm
Option 3: 24 cm
Option 4: 64 cm
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Correct Answer: 48 cm
Solution :
Perpendicular distance =
$p{_1} = \sqrt{3}$
$p{_2} =2\sqrt{3}$
$p{_3} = 5\sqrt{3}$
Height of an equilateral triangle = $\frac{\sqrt{3} \times (a)}{2}$
Height of an equilateral triangle = sum of perpendicular distance with point
The perimeter of an equilateral triangle = $3\times a$
Height of equilateral triangle = sum of perpendicular distance
⇒ $\frac{\sqrt{3} \times (a)}{2} = p{_1} + p{_2}+p{_3}$
⇒ $\frac{\sqrt{3} \times a}{2} = \sqrt{3} +2 \sqrt{3} +5 \sqrt{3}$
⇒ $a = 2(1+2+5)$
⇒ $a= 2\times 8$
⇒ $a = 16\ \mathrm{cm}$
⇒ Perimeter = $3\times16 = 48$ cm
Hence, the correct answer is 48 cm.
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