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Problem of the Week
Problem D
A Circle in a Trapezoid

A circle with centre \(O\) and radius \(15 \text{ m}\) is inside trapezoid \(ABCD\) such that each side of \(ABCD\) is tangent to the circle. In the trapezoid, \(AB \parallel CD\) and \(AD=BC\), so \(ABCD\) is an isosceles trapezoid.

Trapezoid ABCD with side AB shorter than side CD and a circle inside touching each of its sides.

If the area of \(ABCD\) is \(2025 \text{ m}^2\), determine the lengths of \(AD\) and \(BC\).

For this problem, you may want to use the following known results about circles:

  1. If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

  2. A line drawn from the centre of a circle perpendicular to a tangent line meets the tangent line at the point of tangency.