Problem:
Let ABCD be an isoceles trapezoid having parallel bases AB and CD with AB>CD. Line segments from a point inside ABCD to the vertices divide the trapezoid into four triangles whose areas are 2,3,4, and 5 starting with the triangle with base CD and moving clockwise as shown in the diagram below. What is the ratio CDABβ ?
Answer Choices:
A. 3
B. 2+2β
C. 1+6β
D. 23β
E. 32β Solution:
Without the loss of generality, let T have vertices A,B,C, and D, with AB=r and CD=s. Also denote by P the point in the interior of T.
Let X and Y be the feet of the perpendiculars from P to AB and CD, respectively. Observe that PX=r8β and PY=s4β. Now using the formula for the area of a trapezoid yields
