The perpendicular distances of a point inside an equilateral triangle from its three sides are 7 cm, 10 cm and 13 cm. What is the perimeter of the triangle (in cm)?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(60\sqrt{3}\)
Joining the point P to the three vertices splits the triangle into three triangles, each with a side \(a\) as base and one of the distances as height. Their areas add up to the whole area, so the three distances add up to the height of the equilateral triangle.
\(\displaystyle \frac{1}{2}a(7+10+13)=\frac{1}{2}a\times h\), so \(h=30\,\mathrm{cm}\)
Height of an equilateral triangle \(\displaystyle h=\frac{\sqrt{3}}{2}a\)
\(\displaystyle a=\frac{2\times30}{\sqrt{3}}=\frac{60}{\sqrt{3}}=20\sqrt{3}\,\mathrm{cm}\)
Perimeter \(=3a=60\sqrt{3}\,\mathrm{cm}\)
