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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)...")
 
(→‎Problem)
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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]
+
<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 14:14, 20 June 2021

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

$\textbf{(A) }4.75\qquad \textbf{(B) }4.8\qquad \textbf{(C) }5\qquad \textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }3\sqrt{3}$

Solution

$\fbox{B}$

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