From a point in the interior of an equilateral triangle perpendiculars are drawn on 3 sides. The l of perpendiculars = 14 cm, 10 cm and 6 cm. Find area

Question

From a point in the interior of an equilateral triangle perpendiculars are drawn on three sides. The lengths of the perpendiculars are 14 cm, 10 cm and 6 cm. Find the area of the triangle

Answer 1

If you draw lines from the point to each vertex of the triangle, you can find three triangles where the base is the side of the equilateral triangle, and the three perpendiculars are the heights.  Since these three triangles add up to the whole triangle, we can add their areas (let x be the side of the equilateral triangle):

.5(14)(x) + .5 (10)(x) + .5(6)(x) = 15x

 

But we can also just use x to find the area of the overall triangle since the height will be part of a 30-60-90 relationship.  The height of the equilateral triangle is (.5x)√3, and so the area will be .5(x)(.5x)√3 = .25x²√3

 

Now we have two formulas for the area of the triangle which must be equal, so we can use them to solve for x, and then find the area of the triangle.

 

15x = .25x²√3

divide both sides by x, and by .25√3

 

15/(.25√3) = x     (or 60/√3 , or 20√3 if your teacher likes rationalized denominators)

 

Plug x into both area formulas, and you should get the same answer (a bit over 500).