In a right-angled triangle, one of the angles is equal to the arithmetic mean of the other two angles

In a right-angled triangle, one of the angles is equal to the arithmetic mean of the other two angles. find the legs of this triangle if its hypotenuse is C.

By condition, the angle BAC is equal to the arithmetic mean of the angles ABC and ACB.

Angle BAC = (ABC + ACB) / 2 = (90 + ACB) / 2 = 45 + ACB / 2. (1).

The sum of the angles BAC and ACB is 90. (BAC + ACB) = 90.

BAC = 90 – ACB. (2).

Let’s equate expressions 1 and 2.

45 + ACB / 2 = 90 – ACB.

3 * ABC / 2 = 45.

ACB = 2 * 45/3 = 30.

Then the leg BC = AC / 2 = C / 2 cm.

Leg AB = AC * Cos30 = C * √3 / 2 cm.

Answer: The lengths of the legs are C / 2 cm, C * √3 / 2 cm.