# Difference between revisions of "1978 AHSME Problems/Problem 26"

(Created page with "==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)...") |
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==Problem== | ==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 <math>\triangle ABC, AB = 10~ AC = 8</math> and <math>BC = 6</math>. Circle <math>P</math> is the circle with smallest radius which passes through <math>C</math> and is tangent to <math>AB</math>. Let <math>Q</math> and <math>R</math> be the points of intersection, distinct from <math>C</math> , of circle <math>P</math> with sides <math>AC</math> and <math>BC</math>, respectively. The length of segment <math>QR</math> is | In <math>\triangle ABC, AB = 10~ AC = 8</math> and <math>BC = 6</math>. Circle <math>P</math> is the circle with smallest radius which passes through <math>C</math> and is tangent to <math>AB</math>. Let <math>Q</math> and <math>R</math> be the points of intersection, distinct from <math>C</math> , of circle <math>P</math> with sides <math>AC</math> and <math>BC</math>, respectively. The length of segment <math>QR</math> is | ||

<math>\textbf{(A) }4.75\qquad \textbf{(B) }4.8\qquad \textbf{(C) }5\qquad \textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }3\sqrt{3}</math> | <math>\textbf{(A) }4.75\qquad \textbf{(B) }4.8\qquad \textbf{(C) }5\qquad \textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }3\sqrt{3}</math> | ||

+ | |||

==Solution== | ==Solution== | ||

<math>\fbox{B}</math> | <math>\fbox{B}</math> |

## Revision as of 13:14, 20 June 2021

## Problem

In and . Circle is the circle with smallest radius which passes through and is tangent to . Let and be the points of intersection, distinct from , of circle with sides and , respectively. The length of segment is

## Solution