In a rectangular trapezoid ABCD (AD‖BC, AB is perpendicular to CD), the larger base is AD = 15 cm

| September 6, 2021 | education

In a rectangular trapezoid ABCD (AD‖BC, AB is perpendicular to CD), the larger base is AD = 15 cm, the diagonal AC is perpendicular to CD, AC = 12 cm. Find the smaller base of the trapezoid.

By hypothesis, AS is perpendicular to CD, so the triangle ACD is rectangular, then, according to the Pythagorean theorem, CD ^ 2 = AD ^ 2 – AC ^ 2 = 15 ^ 2 – 12 ^ 2 = 225 – 144 = 81.

CD = 9 cm.

From the top of C, we lower the height of CH to the base of the AD.

Let the segment AH = X cm, then DN = (15 – X) cm.

By the Pythagorean theorem in the triangle ACН, CH ^ 2 = AC ^ 2 – AH ^ 2 = 144 – X ^ 2.

By the Pythagorean theorem in the triangle DСН, CH ^ 2 = CH ^ 2 – DН ^ 2 = 81 – (15 – X) ^ 2.

Let’s equate both equalities.

144 – X ^ 2 = 81 – (15 – X) ^ 2.

144 – X ^ 2 = 81 – 225 + 30 * X – X ^ 2.

30 * X = 288.

X = 288/30 = 9.6 cm.

AH = BC = 9.6 cm.

Answer: The smaller base is 9.6 cm.



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