Question

The lengths of the perpendiculars drawn from any point in the interior of an equilateral triangle to the respective sides are p₁, p₂ and p₃. The length of each side of the triangle is

(a) \(\frac{2}{\sqrt3}\) (p₁ + p₂ + p₃) 

(b) \(\frac{1}{3}\) (p₁ + p₂ + p₃) 

(c) \(\frac{1}{\sqrt3}\) (p₁ + p₂ + p₃) 

(d) \(\frac{4}{\sqrt3}\) (p₁ + p₂ + p₃) 

Answer 1

(a) \(\frac{2}{\sqrt3}\) (p₁ + p₂ + p₃) 

Let each side of equilateral ΔPQR = a units. O is any point in the interior of DΔPQR 

⇒ OD = p₁, OE = p₂ and OF = p₃ are perpendiculars on sides PQ, PR and QR respectively. 

∴ Area of ΔPQR

= Area of ΔOPQ + Area of ΔOPR + Area of ΔOQR

= \(\frac{1}{2}\times{a}\times{p}_1+\)\(\frac{1}{2}\times{a}\times{p}_2+ \)\(\frac{1}{2}\times{a}\times{p}_3 \)

= \(\frac{a}{2}(p_1+p_2+p_3)\)

⇒ \(\frac{\sqrt3}{4}a^2\) = \(\frac{a}{2}(p_1+p_2+p_3)\) ⇒ \(\frac{2}{\sqrt3}\) (p₁ + p₂ + p₃)