Difference between revisions of "1994 AHSME Problems/Problem 13"

m (Solution)
Line 1: Line 1:
==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>, then <math>\angle A =</math>
 
<asy>
 
draw((0,0)--(8,0)--(4,12)--cycle);
 
draw((8,0)--(1.6,4.8));
 
label("A", (4,12), N);
 
label("B", (0,0), W);
 
label("C", (8,0), E);
 
label("P", (1.6,4.8), NW);
 
dot((0,0));
 
dot((4,12));
 
dot((8,0));
 
dot((1.6,4.8));
 
</asy>
 
<math> \textbf{(A)}\ 30^{\circ} \qquad\textbf{(B)}\ 36^{\circ} \qquad\textbf{(C)}\ 48^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 72^{\circ} </math>
 
 
==Solution==
 
==Solution==
 
<asy>
 
<asy>

Revision as of 18:54, 7 May 2023

Solution

[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); MA("2x",10,B,P,C,1,3,blue); MA("2x",10,C,B,P,1,3,blue);[/asy]

Let $\angle A=x$. Since $AP=PC$, we have $\angle ACP=x$ as well. Then $\angle APC=180-2x\implies\angle BPC=\angle CBP=2x$. Since $AB=AC$, we have $\angle CBP=\frac{180-x}{2}$.

So $2x=\frac{180-x}{2}\implies 5x=180\implies x=\angle A=\boxed{\textbf{(B) }36^\circ.}$


--Solution by TheMaskedMagician

See Also

1994 AHSME (ProblemsAnswer KeyResources)
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. AMC logo.png