Calculate the lengths of the legs of a right-angled triangle if their projection onto the hypotenuse is 9 cm and 16 cm.

Consider a right-angled triangle ABC with right angle B.

Let’s lower the height BH to the hypotenuse. Therefore, the projection of the legs to the hypotenuse

AH = 9, CH = 16.

Triangles ABH and BCH are similar in three angles, since,

AHB = CHB = 90 °, BAH = CBH, ABH = BCH.

Then:

AH / BH = BH / CH,

BH ^ 2 = AH * CH = 9 * 16,

BH = 3 * 4 = 12.

Then by the Pythagorean theorem for right-angled triangles ABH, CBH we obtain:

AB ^ 2 = BH ^ 2 + AH ^ 2 = 12 ^ 2 + 9 ^ 2 = 15 ^ 2,

AB = 15.

BC ^ 2 = BH ^ 2 + CH ^ 2 = 12 ^ 2 + 16 ^ 2 = 400,

BC = 20.

Answer: the lengths of the legs are 15 and 20.