The two sides of the triangle are equal to 17 cm and 25 cm. The height divides the third side into segments

The two sides of the triangle are equal to 17 cm and 25 cm. The height divides the third side into segments, the difference between which is equal to 12 cm. Find the perimeter of the triangle.

The height drawn to the unknown side divides it into segments equal to the projections of the other two sides of the triangle. Let the smaller segment be equal to x cm, then the larger one is equal to x + 12 cm. Obviously, the segment equal to x is the projection of the side equal to 17 cm, x + 12 is the projection of the side equal to 25 cm.

Thus, we have two right-angled triangles with a common leg equal to the height of the triangle. For each of the right-angled triangles, according to the Pythagorean theorem, we express the height through the other two sides and equate the expressions obtained.

1) h ^ 2 = 17 ^ 2 – x ^ 2;

2) h ^ 2 = 25 ^ 2 – (x + 12) ^ 2.

17 ^ 2 – x ^ 2 = 25 ^ 2 – (x + 12) ^ 2;

289 – x ^ 2 = 625 – x ^ 2 – 24x – 144;

24x = 625 – 289 – 144;

24x = 192;

x = 192/24 = 8 cm.

The unknown side is equal to the sum of the lengths of the segments into which the height drawn to it divides it:

c = x + x + 12 = 28 cm.

The perimeter of a triangle is equal to the sum of the lengths of all its sides:

P = 17 + 25 + 28 = 70 cm.