In a rectangular trapezoid, the bases are 8 and 4. The area of the trapezoid is 24. Find the acute angle

In a rectangular trapezoid, the bases are 8 and 4. The area of the trapezoid is 24. Find the acute angle at the larger base of the trapezoid.

Given: ABCD – trapezoid, ∠BAD = 90 °, BC = 4, AD = 8, SABCD = 24.

Find: ∠ СDА -?

Decision:

Decision:

The area of the trapezoid is: S = 1/2 * (BC + AD) * AB, where AB is the height, because ∠ BAD = 90 ° – according to the condition of the problem. Let us express the height AB from the area formula.

24 = 1/2 * (4 + 8) * AB;

24 = 1/2 * 12 * AB;

48 = 12 * AB;

AB = 4.

Let’s draw the height CK from the top of point C. By the trapezoid property AB = CK = 4.

Consider a triangle CKD: ∠ CKD = 90 °, KD = AD – BC = 8 – 4 = 4. Because CК = КD = 4, then the triangle CKD is isosceles, therefore ∠ КСD = ∠ CDK = 45 °. Therefore ∠ СDA = 45 °.

Answer: ∠ CDA = 45 °.