Point M lies within an angle of 60 degrees. The distance from point M

Point M lies within an angle of 60 degrees. The distance from point M to each side of the angle is 5 cm. Find the distance from point M to the apex of the angle.

Let ∠A be given, MB = MC = 5 cm. It is necessary to find AM.

Since the distance from point M to each side ∠A is the same, it lies on the bisector ∠A. Thus, △ ABM = △ ACM, ∠MAB = ∠MAC = ∠A / 2 = 60 ° / 2 = 30 °.

Consider △ ACM: ∠ACM = 90 ° (since the distance from point M to side ∠A is a perpendicular), ∠MAC = 30 °, AM – hypotenuse, MC = 5 cm and AC – legs.

In a right-angled triangle, the leg, opposite an angle of 30 °, is equal to half the hypotenuse.

The MC leg lies opposite ∠MAC = 30 °, therefore MC = AM / 2.

AM / 2 = 5;

AM = 2 * 5 (proportional);

AM = 10 cm.

Answer: AM = 10 cm.