In a right-angled triangle, the projections of the legs to the hypotenuse are 25 and 16 mm.

In a right-angled triangle, the projections of the legs to the hypotenuse are 25 and 16 mm. Find the legs of this triangle and the height dropped from the top of the right angle and the hypotenuse.

Let us prove that the triangles ABН and BCH are similar.

Let the angle BCH = X0, then the angle BCH = (90 – X) 0.

In triangle ABН, angle ABН = (90 – СВН) = (90 – (90 – X) = X0.

Then the triangles ABН and СВН are similar in acute angle.

Then, in similar triangles:

ВН / AН= CH / ВН.

BH ^ 2 = AH * CH = 25 * 16 = 400.

BH = 20 cm.

In a right-angled triangle ABН, by the Pythagorean theorem, we define the length of the hypotenuse AB.

AB ^ 2 = AH ^ 2 + BH ^ 2 = 625 + 400 = 1025.

AB = 5 * √41 cm.

Similarly, in a right-angled triangle BCH, BC ^ 2 = CH ^ 2 + BH ^ 2 = 256 + 400 = 656.

BC = 4 * √42 cm.