A circle is inscribed in a rectangular trapezoid. Find the area of a trapezoid if its base equals a and b.

univerkov | March 12, 2021 | education

Let’s draw the height of the CH.

The quadrilateral ABCН is a rectangle, then AH = BC = a cm, then DH = AD – AH = (b – a) cm.

Then, by the Pythagorean theorem, СD^2 = h ^ 2 + (a – b) ^ 2.

CD = √ (h ^ 2 + (a – b) ^ 2).

Since a circle is inscribed in the trapezoid, the sum of the opposite sides of the trapezoid is equal to.

BC + AD= AB + CD.

a + b = h + √ (h ^ 2 + (a – b) ^ 2).

(a + b) – h = √ (h ^ 2 + (a – b) ^ 2).

Let’s square both sides.

((a + b) – h) ^ 2 = (√ (h ^ 2 + (a – b) ^ 2)) ^ 2.

a ^ 2 + 2 * a * b + b ^ 2 – 2 * (a + b) * h + h ^ 2 = h ^ 2 + a ^ 2 – 2 * a * b + b ^ 2.

a ^ 2 + 2 * a * b + b ^ 2 – 2 * a * h – 2 * b * h + h ^ 2 = h ^ 2 + a ^ 2 – 2 * a * b + b ^ 2.

2 * h * (a + b) = 4 * a * b.

h = 2 * a * b / (a + b).

Determine the area of the trapezoid.

Savsd = (a + b) * h / 2 = (a + b) * 2 * a * b / (a + b) / 2 = a * b.

Answer: The area of the trapezoid is (a * b).

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