In the trapezoid ABCD AB = CD, the height BH divides the base into two segments

In the trapezoid ABCD AB = CD, the height BH divides the base into two segments, the smaller of which is 5 cm. Find AD if its middle line is 9 cm.

Since, by condition, AB = CD, the trapezoid is isosceles.

By the property of the height of an isosceles trapezoid, the height lowered from the top of an obtuse angle divides the larger base into two segments, the smaller of which is equal to the half-difference of the bases AH = (AD – BC) / 2, and the larger half-sum of the bases HD = (AD + BC) / 2 …

Since (AD + BC) / 2 is the formula for the midline, then НD = КМ = 9 cm.

Then the base АD = АН + НD = 5 + 9 = 14 cm.

Answer: Length АD = 14 cm.