The bisectors of angles A and B at the lateral side AB of the trapezoid ABCD meet

The bisectors of angles A and B at the lateral side AB of the trapezoid ABCD meet at point F. Find AB if AF = 24, BF = 32.

Consider the angles A and B at the lateral side AB of the trapezoid ABCD. According to the theorem, the corresponding angles (<BAD + <ABC) for parallel sides ВС and АD in the sum equal 180 °.

And since AF and BF are the bisectors of angles A and B, then in the formed triangle ABF, the sum of the angles (<ABF + <BAF) = 180 ° / 2 = 90 °. Hence, triangle ABF is rectangular. The AB side in it is the hypotenuse, which is determined by the Pythagorean theorem. Let’s find the AB side:

AB = √ [(AF) ^ 2 + (BF) ^ 2] = √ [(24) ^ 2 + (32) ^ 2] = √ [576 + 1024] = √ (1600) = 40.