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Difference between revisions of "1983 AHSME Problems/Problem 30"

(This may be a controversial choice, but I deleted the section that said, "MATH IS SOOOO FUN" (original person who added it: Piano man123). I know many people believe that it is very crucial for the page, but I think the main solution conveys the message.)
m (→‎Problem: Replaced image with Asy code.)
Line 3: Line 3:
 
Distinct points <math>A</math> and <math>B</math> are on a semicircle with diameter <math>MN</math> and center <math>C</math>.  
 
Distinct points <math>A</math> and <math>B</math> are on a semicircle with diameter <math>MN</math> and center <math>C</math>.  
 
The point <math>P</math> is on <math>CN</math> and <math>\angle CAP = \angle CBP = 10^{\circ}</math>. If <math>\stackrel{\frown}{MA} = 40^{\circ}</math>, then <math>\stackrel{\frown}{BN}</math> equals
 
The point <math>P</math> is on <math>CN</math> and <math>\angle CAP = \angle CBP = 10^{\circ}</math>. If <math>\stackrel{\frown}{MA} = 40^{\circ}</math>, then <math>\stackrel{\frown}{BN}</math> equals
 +
<asy>
 +
import geometry;
 +
import graph;
  
−
[[Image:434fe2d2439016eb468df4ec703fe001e3f9392d.png|center|250px]]
+
unitsize(2 cm);
 +
 
 +
pair A, B, C, M, N, P;
 +
 
 +
M = (-1,0);
 +
N = (1,0);
 +
C = (0,0);
 +
A = dir(140);
 +
B = dir(20);
 +
P = extension(A, A + rotate(10)*(C - A), B, B + rotate(10)*(C - B));
 +
 
 +
draw(M--N);
 +
draw(arc(C,1,0,180));
 +
draw(A--C--B);
 +
draw(A--P--B);
 +
 
 +
label("$A$", A, NW);
 +
label("$B$", B, E);
 +
label("$C$", C, S);
 +
label("$M$", M, SW);
 +
label("$N$", N, SE);
 +
label("$P$", P, S);
 +
</asy>
  
 
<math>\textbf{(A)}\ 10^{\circ}\qquad
 
<math>\textbf{(A)}\ 10^{\circ}\qquad

Revision as of 23:40, 2 December 2021

Problem

Distinct points $A$ and $B$ are on a semicircle with diameter $MN$ and center $C$. The point $P$ is on $CN$ and $\angle CAP = \angle CBP = 10^{\circ}$. If $\stackrel{\frown}{MA} = 40^{\circ}$, then $\stackrel{\frown}{BN}$ equals [asy] import geometry; import graph; unitsize(2 cm); pair A, B, C, M, N, P; M = (-1,0); N = (1,0); C = (0,0); A = dir(140); B = dir(20); P = extension(A, A + rotate(10)*(C - A), B, B + rotate(10)*(C - B)); draw(M--N); draw(arc(C,1,0,180)); draw(A--C--B); draw(A--P--B); label("$A$", A, NW); label("$B$", B, E); label("$C$", C, S); label("$M$", M, SW); label("$N$", N, SE); label("$P$", P, S); [/asy]

$\textbf{(A)}\ 10^{\circ}\qquad \textbf{(B)}\ 15^{\circ}\qquad \textbf{(C)}\ 20^{\circ}\qquad \textbf{(D)}\ 25^{\circ}\qquad \textbf{(E)}\ 30^{\circ}$

Solution

Since $\angle CAP = \angle CBP = 10^\circ$, quadrilateral $ABPC$ is cyclic (as shown below) by the converse of the theorem "angles inscribed in the same arc are equal".

[asy] import geometry; import graph; unitsize(2 cm); pair A, B, C, M, N, P; M = (-1,0); N = (1,0); C = (0,0); A = dir(140); B = dir(20); P = extension(A, A + rotate(10)*(C - A), B, B + rotate(10)*(C - B)); draw(M--N); draw(arc(C,1,0,180)); draw(A--C--B); draw(A--P--B); draw(A--B); draw(circumcircle(A,B,C),dashed); label("$A$", A, W); label("$B$", B, E); label("$C$", C, S); label("$M$", M, SW); label("$N$", N, SE); label("$P$", P, S); [/asy]

Since $\angle ACM = 40^\circ$, $\angle ACP = 140^\circ$, so, using the fact that opposite angles in a cyclic quadrilateral sum to $180^{\circ}$, we have $\angle ABP = 40^\circ$. Hence $\angle ABC = \angle ABP - \angle CBP = 40^ \circ - 10^\circ = 30^\circ$.

Since $CA = CB$, triangle $ABC$ is isosceles, with $\angle BAC = \angle ABC = 30^\circ$. Now, $\angle BAP = \angle BAC - \angle CAP = 30^\circ - 10^\circ = 20^\circ$. Finally, again using the fact that angles inscribed in the same arc are equal, we have $\angle BCP = \angle BAP = \boxed{\textbf{(C)}\ 20^{\circ}}$.

See Also

1983 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 29 		Followed by
Last Problem
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


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