The projections of the legs to the hypotenuse are 2√5 and 8√5. Find the height drawn from the top of the right angle.

Let the angle ∠C be a straight line in a triangle ABC. Then the height CH, lowered from this angle by the hypotenuse AB, divides it into two similar triangles ACH and BCH. Let’s compose the ratio of the sorons of these triangles:

CH / 8√5 = 2√5 / CH,

from where

CH ^ 2 = 2√5 * 8√5;

CH ^ 2 = 16 * 5.

Then the desired length of the height CH is equal to

CH = 4√5.