Question

from point in the interior of an equilateral triangle , perpendicular are drawn on each of its three sides.the length of the perpendiculars are 14cm,10cm and 6cm.find the area of the triangle

Answer 1

Let height from A on the side a be h1, height from B on side b be h2, and height from C on side c be h3
ah1 = bh2 = ch3 = 2 x Area of triangle
Thus a : b : c = (1/h1) : (1/h2) : (1/h3)
Thus a : b : c = (1/14) : (1/10) : (1/6) = (840/14) : (840/10) : (840/6) = 60 : 84 : 140 =
15 : 24 : 35
Thus a = 15k, b = 24k, c = 35k
Semi perimeter = s = (15k + 24k + 35k)/2 = 32k
By Herons Formula area =
$\sqrt{s(s - a)(s - b)(s - c)} = \sqrt{32k.17k.8k.3k} = \sqrt{13056 {k}^{4} }$