In trapezoid ABCD, angle A = 90 degrees, side CD and diagonal AC intersect at right

univerkov | July 25, 2021 | education

In trapezoid ABCD, angle A = 90 degrees, side CD and diagonal AC intersect at right angles, AD = 5 cm, CD = 3 cm. Find the area of the trapezoid.

Since AC is perpendicular to CD, the ACD triangle is rectangular, from which, according to the Pythagorean theorem, we determine the length of the AC leg.

AC ^ 2 = AD ^ 2 – CD ^ 2 = 25 – 9 = 16.

AC = 4 cm.

Let’s draw the height of CH, which is the height of the trapezoid and the height of the triangle ACD.

The area of the triangle ACD is equal to: Sacd = AC * CD / 2 = 4 * 3/2 = 6 cm2.

On the other hand, the area of the ACD triangle is equal to:

Savs = AD * CH / 2.

Then AD * CH / 2 = 6 cm2.

CH = 2 * 6/5 = 12/5 = 2.4 cm.

From a right-angled triangle ACH, AH ^ 2 = AC ^ 2 – CH ^ 2 = 16 – 5.76 = 10.24. AH = BC = 3.2 cm.

Determine the area of the trapezoid.

Savsd = (BC + AD) * CH / 2 = (3.2 + 5) * 2.4 / 2 = 9.84 cm2.

Answer: The area of the trapezoid is 9.84 cm2.

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