Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.8 AMC 10/12 Find the answer Italy

Problem:

A right trapezoid with longer base $AB$ and shorter base $CD$ is circumscribed about a circle of radius $10$ . It is known that the oblique side $BC$ measures $24$ . What is the distance between the midpoints of $BC$ and $AD$ ?

Pick one

Solution

Solution:

The answer is (D). By Thales' Theorem the distance between the midpoints of the oblique sides is given by half the sum of the bases. Since the trapezoid is circumscribed about a circle, the sum of the bases equals the sum of the oblique sides. Since the trapezoid is a right trapezoid, the shorter oblique side is equal to the diameter of the inscribed circle, so the answer is given by $\frac{20+24}{2}=22$ .

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.