In a triangle ABC, the angle is 90 degrees, BD is the height, AB is equal to 2BD. Prove that 3AC is equal to 4AD.

Let’s say BD = x. So AB = 2x.

Let us find AD for the Pythagorean theorem from the triangle BDA.

(2x) ˄2 = x˄2 + АD˄2;

AD˄2 = 3x˄2;

AD = x√3.

Since the hypotenuse AB is twice the leg BD, the angle A = 30 degrees, and the angle C = 60 degrees.

In a triangle ВDC, if the angle С = 60 degrees, then the angle ВDC = 30 degrees. Means 2DС = ВС.

Let’s say DC = y, BC = 2y.

In the BDC triangle:

(2y) ˄2 = x˄2 + y˄2;

x˄2 = 3y˄2;

y˄2 = x˄2 / 3;

y = x / √3.

DC = x / √3, BC = 2x / √3.

3АС = 3 * (x√3 + x / √3) = 4х√3.

4AD = 4x√3.

As you can see, 3АС = 4АD.