Multiple choice

A circle is inscribed in trapezoid $PQRS$. If $PS=QR=25\;cm,\,PQ=18\;cm\;and\;SR=32\;cm$, what is the length of the diameter of the circle ?

  1. $14\;cm$
  2. $25\;cm$
  3. $24\;cm$
  4. $\sqrt{674}\;cm$
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C Correct answer
Explanation

For a circle inscribed in a trapezoid, the height of the trapezoid equals the diameter. Using the properties of an isosceles trapezoid with parallel sides 18 and 32, the base segments are (32-18)/2 = 7. Using Pythagoras on the triangle formed by the side (25) and the base segment (7), height = sqrt(25^2 - 7^2) = sqrt(625 - 49) = sqrt(576) = 24 cm.

AI explanation

For a trapezoid with an inscribed circle, the height can be found using the Pythagorean theorem on the right triangle formed by the height, the leg, and half the difference of the parallel bases. Half the difference of the bases 32 and 18 is 7. Applying the theorem gives the height as the square root of (25 squared minus 7 squared), which is the square root of 576. The height of the trapezoid equals the diameter of the inscribed circle, so the diameter is 24 cm.

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