In an isosceles trapezoid, the lateral side is equal to the half-sum of the bases and is equal to A

In an isosceles trapezoid, the lateral side is equal to the half-sum of the bases and is equal to A; the geometric mean of the bases is P. Find the area of the given trapezoid.

By condition, (AD + BC) / 2 = AB = a cm.

BC * AD = P2.

Let’s build the height BH.

In an isosceles trapezoid, AB = (AD – BC) / 2.

In a right-angled triangle ABN, according to the Pythagorean theorem:

BH ^ 2 = AB ^ 2 – AH ^ 2 = (AA + BC) ^ 2/4 – (AA – BC) ^ 2/4 = (AA ^ 2 + 2 * AA * BC + BC ^ 2 – AA ^ 2 + 2 * AA * BC – BC ^ 2) / 4 = 4 * AA * BC / 4 = P2.

BH = R.

Then Savsd = (AD + BC) * BH / 2 = a * P cm2.

Answer: The area of the trapezoid is equal to a * P cm2.