The lengths of the sides of the triangle are in relation to 15: 22: 15, connecting the midpoints of the sides

The lengths of the sides of the triangle are in relation to 15: 22: 15, connecting the midpoints of the sides, we get a triangle with an area of 22 √ 26. Find the perimeter of the original triangle.

The sides of the triangle are related as 15: 22: 15, let one side be equal to 15x, then the other two will be equal to 22x and 15x.

Since the vertices of the small triangle connect the midpoints of the sides of the large triangle, each side of the small triangle is the midline of the large triangle (by the property of the midline) and is equal to half of the side lying opposite. Therefore, the sides of the small triangle are 7.5x, 11x, and 7.5x.

Let us express the area of ​​a small triangle using Heron’s formula:

S = √p (p – a) (p – b) (p – c) (p is the semiperimeter of the triangle, a, b and c are the sides of the triangle).

p = (7.5x + 11x + 7.5x): 2 = 13x.

S = √ (13x * (13x – 7.5x) * (13x – 11x) * (13x – 7.5x)) = √ (13x * 5.5x * 2x * 5.5x) = 5.5x²√26.

Since the area is 22√26, the equation is:

5.5x²√26 = 22√26.

5.5x² = 22.

x² = 22: 5.5 = 220: 55 = 4.

x = 2.

This means that the sides of the big triangle are equal:

15x = 15 * 2 = 30,

22x = 22 * ​​2 = 44.

15x = 30.

Therefore, the perimeter is:

P = 30 + 44 + 30 = 104.