The lengths of the perpendiculars drawn from any point in the interior of an equilateral -Maths 9th

The lengths of the perpendiculars drawn from any point in the interior of an equilateral -Maths 9th
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(a) \(rac{2}{\sqrt3}\) (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= \(rac{1}{2} imes{a} imes{p}_1+\)\(rac{1}{2} imes{a} imes{p}_2+ \)\(rac{1}{2} imes{a} imes{p}_3 \)= \(rac{a}{2}(p_1+p_2+p_3)\)⇒ \(rac{\sqrt3}{4}a^2\) = \(rac{a}{2}(p_1+p_2+p_3)\) ⇒ \(rac{2}{\sqrt3}\) (p1 + p2 + p3)
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