Difference between revisions of "1994 AHSME Problems/Problem 13"
(Created page with "==Problem== In triangle <math>ABC</math>, <math>AB=AC</math>. If there is a point <math>P</math> strictly between <math>A</math> and <math>B</math> such that <math>AP=PC=CB</math...") |
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− | == | + | ==Solution== |
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<asy> | <asy> | ||
− | + | import cse5; | |
− | + | pathpen=black; | |
− | + | pointpen=black; | |
− | + | pair A=(4,12),B=(0,0),C=(8,0),P=(1.6,4.8); | |
− | + | D(MP("A",A,N)--MP("B",B,W)--MP("C",C,E)--cycle); | |
− | + | D(C--MP("P",P,NW)); | |
− | + | D(A);D(B);D(C);D(P); | |
− | + | MA("x",10,B,A,C,1,1); | |
− | + | MA("x",10,A,C,P,1,1); | |
− | + | MA(-20,"180-2x",8,C,P,A,1,2); | |
− | </asy> | + | MA("2x",10,B,P,C,1,3,blue); |
− | <math> \ | + | MA("2x",10,C,B,P,1,3,blue);</asy> |
− | == | + | |
+ | Let <math>\angle A=x</math>. Since <math>AP=PC</math>, we have <math>\angle ACP=x</math> as well. Then <math>\angle APC=180-2x\implies\angle BPC=\angle CBP=2x</math>. Since <math>AB=AC</math>, we have <math>\angle CBP=\frac{180-x}{2}</math>. | ||
+ | |||
+ | So <math>2x=\frac{180-x}{2}\implies 5x=180\implies x=\angle A=\boxed{\textbf{(B) }36^\circ.}</math> | ||
+ | |||
+ | |||
+ | --Solution by [http://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=200685 TheMaskedMagician] | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{AHSME box|year=1994|num-b=12|num-a=14}} | ||
+ | {{MAA Notice}} |
Latest revision as of 19:54, 7 May 2023
Solution
Let . Since , we have as well. Then . Since , we have .
So
--Solution by TheMaskedMagician
See Also
1994 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.