Given a right-angled triangle, one of the angles is 60 degrees, the difference between the hypotenuse

Given a right-angled triangle, one of the angles is 60 degrees, the difference between the hypotenuse and the smaller leg is 3 cm, find the hypotenuse and the smaller leg.

Let us introduce the notation: c – hypotenuse, b – smaller leg.

By condition: c – b = 3.

It is known that in a triangle there is a larger angle opposite the larger side. An angle of 60 ° is the larger of the two acute angles of this triangle, which means that the smaller leg is adjacent to an angle of 60 °.

The ratio of the adjacent leg to the hypotenuse is equal to the cosine of the angle, which means:

cos 60 ° = b / c.

Hence, c = b / cos 60 ° = b / 0.5 = 2b.

Thus, c – b = 2b – b = 3, hence:

b = 3 cm – smaller leg;

c = 2 * 3 = 6 cm – hypotenuse.