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.
We will express this as an equation h2 = 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 ⇔ h2 = xy ⇔ h = √xy
Furthermore, we have a2 = yc and b2 = 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, (a2 + b2 = c2)
(15)2 + b2 = (17)2
(15)2 + b2 = (17)2
⇒225 + b2 = 289
⇒ b2 = 64
⇒ b = sqrt(64) = 8