Thread: Given the altitude:

The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into segments such that their lengths are in the ratio of 1:4. If the length of the altitude is 8, find the length of: (a) each segment of the hypotenuse and (b) the longer leg of the triangle.

Let L be the hyp of the triangle . Its area = 0.5 x L x 8 = 4L --- (1)

The smaller triangle formed has a base of (1/5) L with a height 8 .

Its third side = [ 8^2 + L^2 / 5 ^2] ^ 0.5 .

The larger of the two triangles formed has a base of (4/5)L and height of 8.

Its third side = [ 8^2 + 16L^2 / 25] ^ 0.5

The area of the triangle formed by

[ 8^2 + L^2 / 5 ^2] ^ 0.5 , [ 8^2 + 16L^2 / 25] ^ 0.5 and L is

0.5 x [ 8^2 + L^2 / 5 ^2] ^ 0.5 x [ 8^2 + 16L^2 / 25] ^ 0.5 = 4L

from (1) as both are the same triangle. Solve for L and find

1/5 L , 4/5 L and , [ 8^2 + 16L^2 / 25] ^ 0.5