The lesser base of an isosceles trapezoid is 7. The height of the trapezoid is 10.

The lesser base of an isosceles trapezoid is 7. The height of the trapezoid is 10. The tangent of an acute angle is 4. Find the midline of the trapezoid.

Let’s build the height of the HВ.

In the right-angled triangle ABН, we determine the length of the leg AH.

tgBAH = BH / AH.

AH = BH / tgBAH = 10/4 = 2.5 cm.

Let’s build the height of the CК.

Since the trapezoid is isosceles, the right-angled triangles AВН and СDK are equal in hypotenuse and acute angle, then DK = AH = 2.5 cm.

НK = ВС = 7 cm, since ВСKН is a rectangle, then ВD = AH + NK + DK = 2.5 + 7 + 2.5 = 12 cm.

Let’s define the middle line of the trapezoid.

РM = (BC + AD) / 2 = (7 + 12) / 2 = 19/2 = 9.5 cm.

Answer: The middle line of the trapezoid is 9.5 cm.