Find the length of the altitude to the hypotenuse if a right triangle whose sides have lengths 5,12, and 13.

Question

Find the length of the altitude to the hypotenuse if a right triangle whose sides have lengths 5,12, and 13.

Answer 1

the altitude to the hypotenuse of a right triangle is the geometric mean of the two segments of the hypotenuse

draw the triangle ABC where A is at the top, B is at the bottom left and C is at the bottom right

<ABC is a right angle

draw the altitude to the hypotenuse and call this point on the hypotenuse point D

call the two segments of the hypotenuse x and 13-x

from the above theorem...

AD²=x(13-x)

using the smaller triangle on the left where 5 is the hypotenuse, x is one leg and the altitude, AD, is the other leg

5²=x²+AD²

substitute for AD²

5²=x²+x(13-x)

25=x²+13x-x²

the x²'s will cancel

25=13x

x=25/13

x=1.923

now use the Pythagorean Theorem again to solve for the altitude AD

5²=(1.923)²+AD²

25=3.6979+AD²

25-3.6979=AD²

AD²=21.3021

take the square root of both sides

AD=√21.3021

AD=4.615419 is the altitude to the hypotenuse of the right triangle

Answer 2

Draw the situation, let the hypotenuse be divided into two pieces, y and 13-y, by that altitude to it, which will have length x.

 

Label the angle whose sine is x/5 as θ; in the other small rt. triangle, there's another angle also called θ and it has sine y/12.

 

x/5 = y/12 and also x² + y² = 12²

 

From these two equations you can solve for x.