In an isosceles trapezoid ABCD, the bases of BC and AD are 6 and 12, respectively.

In an isosceles trapezoid ABCD, the bases of BC and AD are 6 and 12, respectively. Find the length of the height BH if the side length is 5.

Let us lower the heights BH and CK to the larger base of the trapezoid ABCD.

The quadrilateral BНKC is a rectangle, then HK = BC = 6 cm, and the triangles ABН and CDK are rectangular.

In triangles ABН and CDK, the hypotenuse AB and CD are equal, since the trapezoid is isosceles, and, accordingly, the angle BAН = CDK, then the triangles are equal in hypotenuse and acute angle, which means AH = DK = (AD – НK) / 2 = (12 – 6 ) / 2 = 3 cm.

In a right-angled triangle ABН, BH ^ 2 = AB ^ 2 – AH ^ 2 = 25 – 9 = 16.

BH = 4 cm.

Answer: The height of the trapezoid is 4 cm.