The angles of a triangle are 1: 1: 2 and the longest side is 13. Find the height drawn to this side.

Let the angle BAC = X0, the angle ACB = X0, then the angle ABC = 2 * X0.

The sum of the interior angles of the triangle is 180, then (X + X + 2 * X) = 180.

4 * X = 180.

X = 180/4 = 45.

Angle BAC = ACB = 45.

Angle ABC = 2 * 45 = 90.

Then the triangle ABC is rectangular and isosceles, and the height BH is also the median of the triangle ABC. Then AH = CH = AC / 2 = 13/2 = 7.5 cm.

In a right-angled triangle ABН, the angle BAН = 45, then the legs of this triangle are equal.

BH = AH = 7.5 cm.

Answer: The height of the triangle is 7.5 cm.