SOLUTION: The altitude of the hypotenuse of a right triangle divides the hypotenuse into 45in. And 5 in segments. What is the length of the altitude?

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Question 955928: The altitude of the hypotenuse of a right triangle divides the hypotenuse into 45in. And 5 in segments. What is the length of the altitude?
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
A theorem about this triangle situation has already been established. One may also be able to apply Pythagorean Theorem to form three equations for a system about this triangle.

Make the described drawing:
Right Triangle with the hypotenuse on bottom as a base and the right angle at the top. Show the altitude and label it as a. Label the left portion of the base hypotenuse as 45, and label the right portion of the base as 5. Label the left leg as s, and label the right leg as t. The altitude divides the original triangle into two smaller right triangles.

Pythagorean Theorem applied to each of the THREE triangles will give
system%2845%5E2%2Ba%5E2=s%5E2%2C5%5E2%2Ba%5E2=t%5E2%2Cs%5E2%2Bt%5E2=%2845%2B5%29%5E2%29

system%2845%5E2%2Ba%5E2=s%5E2%2C5%5E2%2Ba%5E2=t%5E2%2Cs%5E2%2Bt%5E2=50%5E2%29

system%28a%5E2-s%5E2=-45%5E2%2Ca%5E2-t%5E2=5%5E2%2Cs%5E2%2Bt%5E2=50%5E2%29

Taking R2, solving for t^2, and substituting into R3,
s%5E2%2Ba%5E2=2525
a%5E2%2Bs%5E2=2525

Simpler system now is
system%28a%5E2-s%5E2=-45%5E2%2Ca%5E2%2Bs%5E2=2525%29.
This system is ready for easy elimination of s^2, just through adding the corresponding members.

a%5E2-s%5E2%2Ba%5E2%2Bs%5E2=-45%5E2%2B2525
2a%5E2=2525-45%5E2
2a%5E2=2025-2025
2a%5E2=500
highlight%28a=5%2Asqrt%2810%29%29

Might be convenient, learn the rule from Geometry. Maybe simpler to solve.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

The altitude of the hypotenuse of a right triangle divides the hypotenuse into 45in. And 5 in segments. What is the length of the altitude?
Length of altitude that divides hypotenuse into 45" and 5" segments: highlight_green%2815%29 inches