In the trapezoid ABCD CD = 7.5; BC = 4; AD = 8.5. Angles A and B are straight. Find AB.

Draw a trapezoid and drop the perpendicular CE from the vertex C to the AD side.

Thus, the trapezoid is divided into a rectangle ABCE with AB = CE and BC = AE, and into a right-angled triangle CED with ∠CED = 90 °.

Find ED:

ED = AD – AE.

Since BC = AE, then

ED = AD – BC;

ED = 8.5-4;

ED = 4.5.

From △ CED, by the Pythagorean theorem, we find CE:

CE ^ 2 + ED ^ 2 = CD ^ 2;

CE ^ 2 = CD ^ 2 – ED ^ 2;

CE ^ 2 = 7.5 ^ 2 – 4.5 ^ 2;

CE ^ 2 = 56.25 – 20.25;

CE ^ 2 = 36;

CE = √36;

CE = 6.

Since in the rectangle ABCE AB = CE, then:

AB = CE = 6.

Answer: AB = 6.