Point M lies inside an equilateral triangle with side 8√3. It is known that the distance from point M

Point M lies inside an equilateral triangle with side 8√3. It is known that the distance from point M to two sides of a given triangle is 5. At what distance does point M lie from the third side?

Since the triangle ABC, by condition, is equilateral, then all its internal angles are equal to 60.

Determine the area of the triangle ABC.

Savs = AB * BC * Sin60 / 2 = 8 * √3 * 8 √3 * √3 / 4 = 48 * √3 cm2.

Also Savs = ВН * AC / 2.

ВН = 2 * Savs / AC = 2 * 48 * √3 / 8 * √3 = 12 cm.

Since MK = MR = 5 cm, the point M lies on the bisector of the angle ABC, then the angle KВM = ABC / 2 = 60/2 = 30.

In a right-angled KMВ triangle, the KM leg lies opposite an angle of 30, then MВ = MK * 2 = 5 * 2 = 10 cm.

MН = ВН – MВ = 12 – 10 = 2 cm.

Answer: From point M to the third side 2 cm.