Question

Suppose that triangle ABC is an equilateral triangle and that P is a point in the interior of this triangle. Prove that the sum of the perpendicular distances from P to each of the sides of the triangle is equal to the height of the triangle.

Answer 1

Step-by-step explanation:

Let P be the interior of triangle ABC

Area of ΔABC = Area of Δ APB + Area of ΔBPC + Area of ΔAPC

1/2 x base x height = ( 1/2 x AB x CP )+( 1/2 x BC x EP )+( 1/2 x AC x PD )

1/2 x base x height = ( 1/2 x AB x CP )+( 1/2 x AB x EP )+( 1/2 x AB x PD )

( since AB =BC =CA in an equilateral triangle)

1/2 x AB x height = 1/2 x AB x ( CP+EP+PD)

1/2 x AB x height = 1/2 x AB x ( sum of perpendicular distances from P to each side of a triangle ).

Thus, the height of an equilateral triangle is equal to the sum of the perpendicular distance from point P to each side of the triangle.

Hence proved.