# The sides of the trapezoid are 15 cm and 20 cm, and the difference between their bases is 25 cm

**The sides of the trapezoid are 15 cm and 20 cm, and the difference between their bases is 25 cm. Find the height of the trapezoid**

Through the vertex B of the trapezoid, draw a straight line BK parallel to CD, then BK = CD = 20 cm.

Since BСВК is a parallelogram, then КВ = ВС, and AK = АD – КD, which by condition is equal to 25 cm.

AK = 25 cm.

The AВK triangle is rectangular, since the Pythagorean theorem is fulfilled in it.

25 ^ 2 = 625.

15 ^ 2 + 20 ^ 2 = 225 + 400 = 625.

Then the BH height is drawn from a right angle to the hypotenuse, which means BH ^ 2 = AH * KH.

Let the length of the segment AH = X cm, then KH = (25 – X).

Let’s define BH in two right-angled triangles, ABH and BKН.

BH ^ 2 = AB ^ 2 – AH ^ 2 = 225 – X ^ 2.

BH ^ 2 = BK ^ 2 – KH ^ 2 = 400 – (25 – X) ^ 2 = 400 – (625 – 50 * X + X ^ 2) = 50 * X – X ^ 2 – 225.

Then 225 – X ^ 2 = 50 * X – X ^ 2 – 225.

50 * X = 450.

X = 450/50 = 9 cm.

Then AH = 9 cm.

BH ^ 2 = 225 – 81 = 144.

BH = 12 cm.

Answer: The height of the trapezoid is 12 cm.