Question

in an equilateral triangle prove that three times the square of one side is equal to the four times the square of one of its altitudes ques.no.16

Answer 1

See below.

Explanation:

Let the side have a length x units .

The triangle formed by an altitude is a 30-69-90 triangle so the sides are in the ratio 2:1:√3 where √3 is the altitude. Therefore if the side of the triangle = x then altitude = √3 * x / 2 = √3x/2.

So we need to prove that 3x^2 = 4 (√3x/2)^2.

Simplifying:-

4 (√3x/2)^2 = 4 *( 3x^2/ 4) = 3x^2. This completes the proof,