Find a leg of a right-angled triangle lying opposite an angle of 60 degrees if its hypotenuse is 8 m.

Given:

right-angled triangle ABC;

angle C = 90 degrees;

angle B = 60 degrees;

AB – hypotenuse,

AB = 8 meters.

Find the length of the AC leg -?

Decision:

Consider a right-angled triangle ABC.

Angle A = 180 – angle B – angle C;

angle A = 180 – 60 – 90;

angle A = 60 degrees.

In a right-angled triangle opposite an angle of 30 degrees lies a leg, which is half the hypotenuse. Then BC = 1/2 * AB = 1/2 * 8 = 4 (meters).

By the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):

AC ^ 2 + BC ^ 2 = AB ^ 2;

AC ^ 2 = AB ^ 2 – BC ^ 2;

AC ^ 2 = 64 – 16;

AC ^ 2 = 48;

AC = 4√3 meters.

Answer: 4√3 meters.