# The hypotenuse of a right-angled triangle is 13, one of the legs is 5. Find the elements of the triangle.

Let’s define the second leg of a right-angled triangle by the Pythagorean theorem.

AB ^ 2 = BC ^ 2 – AC ^ 2.

AB ^ 2 = 13 ^ 2 – 5 ^ 2 = 169 – 25 = 144.

AB = 12 cm.

The leg of a right-angled triangle is equal to the square root of the product of the hypotenuse and the projection of this leg to the hypotenuse.

AC = √ (BC * CH).

AC2 = BC * CH.

CH = 52/13 = 25/13 = 1 (12/13) cm.

AB ^ 2 = BC * BH.

BH = 144/13 = 11 (1/13) cm.

AH = √ (BH * CH) = √ (25/13 * 144/13) = 60/13 = 4 (8/13) cm.

The median AM of the ABC triangle is equal to half the length of the hypotenuse.

AM = BC / 2 = 13/2 = 6.5 cm.

Cos ACB = AC / BC = 5/13.

Angle ACB Arccos 5/13.

Cos ABC = AB / BC = 12/13.

Angle ABC Arccos 12/13.

Answer: AB = 12, AH = 4 (8/13), CH = 1 (12/13), BH = 11 (1/13), AM = 6.5.