Problem 4

In the rectangle ABCD, the point X on AB is chosen so that XB/AX = 2. If is a bisector of , with U and V on side and as shown, compute the ratio PV/UP, where P is the midpoint of . [Problem submitted by Iris Magee, LACC Associate Professor of Mathematics.]

Solution:

Draw a line segment ef through P such that ef is parallel to AB.

Since XB/AX=2, let AX=1, then XB=2. AB=AX+XB=1+2=3. Therefore ef=AB=3.

DeDP ~ DADX, therefore eP/AX=DP/DX=1/2. Therefore eP=1/2, since AX=1.

Pf=ef-eP=3-1/2=5/2. Therefore Pf/eP=(5/2)/(1/2)=5.

DPUe ~ DPVf, therefore PV/UP=Pf/eP=5.