The perimeter of an equilateral triangle is 144 cm, point m is equidistant

The perimeter of an equilateral triangle is 144 cm, point m is equidistant from each side of this triangle by 19 cm, find the distance from point m to the plane of the triangle.

If the perimeter of an equilateral triangle is 144 cm, then each of its sides is equal to:

a = 144/3 = 48 cm.

Point M, equidistant from all sides of a given triangle, is the apex of a regular triangular pyramid, the base of which is an equilateral triangle given by condition. The distance from point M to the plane of the triangle is the height of the pyramid, the distance from point M to the sides of the triangle is the apothem of equal lateral faces.

The height of the pyramid h, the apothem of the lateral facet l and the projection of the apothem of the lateral facet form a right-angled triangle. The projection of the apothem of the side face coincides with the radius of the inscribed circle r, which can be found by the formula r = a / 2√3, where a is the side of the base.

r = 48 / 2√3 = 24 / √3.

By the Pythagorean theorem:

h ^ 2 = l ^ 2 – r ^ 2 = 19 ^ 2 – (24 / √3) ^ 2 = 361 – 576/3 = 361 – 192 = 169 = 132;

h = 13 – distance from point M to the plane of the triangle.