Difference between revisions of "1964 AHSME Problems/Problem 39"

(Created page with "==Problem== The magnitudes of the sides of triangle <math>ABC</math> are <math>a</math>, <math>b</math>, and <math>c</math>, as shown, with <math>c\le b\le a</math>. Through int...")
 
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label("$c$",Z,dir(110));
 
label("$c$",Z,dir(110));
 
</asy>
 
</asy>
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==See Also==
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{{AHSME 40p box|year=1964|num-b=38|num-a=40}}
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[[Category:Introductory Algebra Problems]]
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{{MAA Notice}}

Revision as of 23:19, 24 July 2019

Problem

The magnitudes of the sides of triangle $ABC$ are $a$, $b$, and $c$, as shown, with $c\le b\le a$. Through interior point $P$ and the vertices $A$, $B$, $C$, lines are drawn meeting the opposite sides in $A'$, $B'$, $C'$, respectively. Let $s=AA'+BB'+CC'$. Then, for all positions of point $P$, $s$ is less than:

$\textbf{(A) }2a+b\qquad\textbf{(B) }2a+c\qquad\textbf{(C) }2b+c\qquad\textbf{(D) }a+2b\qquad \textbf{(E) }$ $a+b+c$


[asy] import math;  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("$A$",A,dir(210)); label("$B$",B,dir(90)); label("$C$",C,dir(-30)); label("$A'$",X,dir(-100)); label("$B'$",Y,dir(65)); label("$C'$",Z,dir(20)); label("$P$",P,dir(70)); label("$a$",X,dir(80)); label("$b$",Y,dir(-90)); label("$c$",Z,dir(110)); [/asy]

See Also

1964 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 38
Followed by
Problem 40
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All AHSME Problems and Solutions

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