The perimeter of an isosceles triangle is 48 cm. Through the middle of the height of the triangle

The perimeter of an isosceles triangle is 48 cm. Through the middle of the height of the triangle, lowered to its base, a straight line is drawn parallel to the lateral side. find the perimeter of the triangle that this line cuts off from the given one.

1. By Thales’s theorem, parallel lines MN and BC intersecting the sides of the angle BHC cut off proportional segments from these sides:

CN: HN = BO: HO = 1, hence:

СN = HN = 1/2 * HC. (one)

2. The height BH of an isosceles triangle ABC is also the median:

AH = CH = 1/2 * AC. (2)

3. Triangles AMN and ABC are homothetic with the homothety coefficient:

k = AN: AC = 3HN: 4HN = 3/4.

4. The ratio of the perimeters of homothetic figures is equal to the coefficient of homothety:

P (AMN) / P (ABC) = 3/4;
P (AMN) = 3/4 * P (ABC) = 3/4 * 48 = 36 (cm).
Answer: 36 cm.