# How to find the altitude of a right triangle with only the hypotenuse?

The ratio in which the altitude divides the hypotenuse is always in geometric progression with the altitude.

## Answer: The altitude can be easily found out with help of hypotenuse as per the mentioned formula in the explanation.

We will use the similarity of triangles to answer this question.

**Explanation:**

The measure of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.

"Add the image"

We will express this as an equation h^{2 }= xy

Note also that all the right triangles are similar to each other because of the AA postulate, so △ACB ∼ △ADC ∼ △CDB

Because of the similarity, we get the following ratio and its algebraic rearrangement yields the theorem: hx = yh ⇔ h^{2} = xy ⇔ h = √xy

Furthermore, we have a^{2} = yc and b^{2} = xc

Example:

Given: c = 17, a = 15, what are b, x, y and h?

b is easy, which can be arrived at using the Pythagorean theorem, (a^{2} + b^{2} = c^{2})

(15)^{2} + b^{2} = (17)^{2}

(15)^{2} + b^{2} = (17)^{2}

⇒225 + b^{2} = 289

⇒ b^{2} = 64

⇒ b = sqrt(64) = 8