# 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.

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