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Find the length of QR in circle P

Source: 1978 AHSME Problem 26

June 18, 2014
Asymptoteanalytic geometrygeometryrectangleAMC

Problem Statement

[asy] import cse5; size(180); real a=4, b=3; pathpen=black; pair A=(a,0), B=(0,b), C=(0,0); D(MP("A",A)--MP("B",B,N)--MP("C",C,SW)--cycle); pair X=IP(B--A,(0,0)--(b,a)); D(CP((X+C)/2,C)); D(MP("R",IP(CP((X+C)/2,C),B--C),NW)--MP("Q",IP(CP((X+C)/2,C),A--C+(0.1,0)))); //Credit to chezbgone2 for the diagram[/asy] In ABC\triangle ABC, AB=10 AC=8AB = 10~ AC = 8 and BC=6BC = 6. Circle PP is the circle with smallest radius which passes through CC and is tangent to ABAB. Let QQ and RR be the points of intersection, distinct from CC , of circle PP with sides ACAC and BCBC, respectively. The length of segment QRQR is
<spanclass=latexbold>(A)</span>4.75<spanclass=latexbold>(B)</span>4.8<spanclass=latexbold>(C)</span>5<spanclass=latexbold>(D)</span>42<spanclass=latexbold>(E)</span>33<span class='latex-bold'>(A) </span>4.75\qquad<span class='latex-bold'>(B) </span>4.8\qquad<span class='latex-bold'>(C) </span>5\qquad<span class='latex-bold'>(D) </span>4\sqrt{2}\qquad <span class='latex-bold'>(E) </span>3\sqrt{3}