1978 AHSME Problems/Problem 26

Problem

[asy] size(100); real a=4, b=3; // import cse5; 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 $\triangle ABC, AB = 10~ AC = 8$ and $BC = 6$ . Circle $P$ is the circle with smallest radius which passes through $C$ and is tangent to $AB$ . Let $Q$ and $R$ be the points of intersection, distinct from $C$ , of circle $P$ with sides $AC$ and $BC$ , respectively. The length of segment $QR$ is