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Ratios

Source: 1966 AHSME #32

August 31, 2011
ratiogeometryAMC

Problem Statement

Let MM be the midpoint of side ABAB of the triangle ABCABC. LetPP be a point on ABAB between AA and MM, and let MDMD be drawn parallel to PCPC and intersecting BCBC at DD. If the ratio of the area of the triangle BPDBPD to that of triangle ABCABC is denoted by rr, then
(A) 12<r<1 depending upon the position of P(B) r=12 independent of the position of P(C) 12≤r<1 depending upon the position of P(D) 13<r<23 depending upon the position of P(E) r=13 independent of the position of P\text{(A)}\ \tfrac{1}{2}<r<1\text{ depending upon the position of }P \qquad\\ \text{(B)}\ r=\tfrac{1}{2}\text{ independent of the position of }P\qquad\\ \text{(C)}\ \tfrac{1}{2}\le r<1\text{ depending upon the position of }P \qquad\\ \text{(D)}\ \tfrac{1}{3}<r<\tfrac{2}{3}\text{ depending upon the position of }P \qquad\\ \text{(E)}\ r=\tfrac{1}{3} \text{ independent of the position of }P
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