A circle with center O is inscribed in an isosceles trapezoid ABCD with bases

A circle with center O is inscribed in an isosceles trapezoid ABCD with bases BC = 9 cm and AD = 25 cm. Prove that triangle AOB is right-angled.

By the property of a circle inscribed in a trapezoid, the center of this circle is the point of intersection of the bisectors of the corners of the trapezoid. Then the segments OA and OB are the bisectors of the angles ABC and ВAD.

The sum of the angles at the lateral sides of the trapezoid is 1800. Angle ABC + ВAD = 180, then (ABC + ВAD) / 2 = 90.

Angle ABO + BAO = 900, then angle AOB = 180 – ABO – BAO = 180 – 90 = 90.

Triangle ABO is rectangular with right angle AOB, as required.