MathDB

39

Part of 1964 AMC 12/AHSME

Problems(1)

Magnitudes of triangle ABC

Source: 1964 AHSME Problem 39

3/19/2014
The magnitudes of the sides of triangle ABCABC are aa, bb, and cc, as shown, with c≤b≤ac\le b\le a. Through interior point PP and the vertices AA, BB, CC, lines are drawn meeting the opposite sides in A′A', B′B', C′C', respectively. Let s=AA′+BB′+CC′s=AA'+BB'+CC'. Then, for all positions of point PP, ss is less than:
<spanclass=′latex−bold′>(A)</span>2a+b<spanclass=′latex−bold′>(B)</span>2a+c<spanclass=′latex−bold′>(C)</span>2b+c<spanclass=′latex−bold′>(D)</span>a+2b<spanclass=′latex−bold′>(E)</span><span class='latex-bold'>(A) </span>2a+b\qquad<span class='latex-bold'>(B) </span>2a+c\qquad<span class='latex-bold'>(C) </span>2b+c\qquad<span class='latex-bold'>(D) </span>a+2b\qquad <span class='latex-bold'>(E) </span> a+b+ca+b+c
[asy] import math; defaultpen(fontsize(11pt)); pair A = (0,0), B = (1,3), C = (5,0), P = (1.5,1);
pair X = extension(B,C,A,P), Y = extension(A,C,B,P), Z = extension(A,B,C,P);
draw(A--B--C--cycle); draw(A--X); draw(B--Y); draw(C--Z); dot(P); dot(A); dot(B); dot(C); label("AA",A,dir(210)); label("BB",B,dir(90)); label("CC",C,dir(-30)); label("A′A'",X,dir(-100)); label("B′B'",Y,dir(65)); label("C′C'",Z,dir(20)); label("PP",P,dir(70)); label("aa",X,dir(80)); label("bb",Y,dir(-90)); label("cc",Z,dir(110)); //Credit to bobthesmartypants for the diagram [/asy]
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