# 1955 AHSME Problems/Problem 29

## Problem

In the figure, $PA$ is tangent to semicircle $SAR$; $PB$ is tangent to semicircle $RBT$; $SRT$ is a straight line; the arcs are indicated in the figure. $\angle APB$ is measured by:

$[asy] unitsize(1.2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; pair O1=(0,0), O2=(3,0), Sp=(-2,0), R=(2,0), T=(4,0); pair A=O1+2*dir(60), B=O2+dir(85); pair Pa=rotate(90,A)*O1, Pb=rotate(-90,B)*O2; pair P=extension(A,Pa,B,Pb); pair[] dots={Sp,R,T,A,B,P}; draw(P--P+5*(A-P)); draw(P--P+5*(B-P)); clip((-2,0)--(-2,2.5)--(4,2.5)--(4,0)--cycle); draw(Arc(O1,2,0,180)--cycle); draw(Arc(O2,1,0,180)--cycle); dot(dots); label("S",Sp,S); label("R",R,S); label("T",T,S); label("A",A,NE); label("B",B,N); label("P",P,NNE); label("a",midpoint(Arc(O1,2,0,60)),SW); label("b",midpoint(Arc(O2,1,85,180)),SE); label("c",midpoint(Arc(O1,2,60,180)),SE); label("d",midpoint(Arc(O2,1,0,85)),SW);[/asy]$

$\textbf{(A)}\ \frac{1}{2}(a-b)\qquad\textbf{(B)}\ \frac{1}{2}(a+b)\qquad\textbf{(C)}\ (c-a)-(d-b)\qquad\textbf{(D)}\ a-b\qquad\textbf{(E)}\ a+b$

## Solution

$[asy] unitsize(1.2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; pair O1=(0,0), O2=(3,0), Sp=(-2,0), R=(2,0), T=(4,0); pair A=O1+2*dir(60), B=O2+dir(85); pair Pa=rotate(90,A)*O1, Pb=rotate(-90,B)*O2; pair P=extension(A,Pa,B,Pb); pair[] dots={Sp,R,T,A,B,P}; draw(P--P+5*(A-P)); draw(P--P+5*(B-P)); clip((-2,0)--(-2,2.5)--(4,2.5)--(4,0)--cycle); draw(Arc(O1,2,0,180)--cycle); draw(Arc(O2,1,0,180)--cycle); draw(P--R, linetype("8 8")); draw(A--O1, linetype("8 8")); draw(B--O2, linetype("8 8")); draw(P--O1, linetype("8 8")); draw(P--O2, linetype("8 8")); dot(dots); label("S",Sp,S); label("R",R,S); label("T",T,S); label("A",A,NE); label("B",B,N); label("P",P,NNE); label("a",midpoint(Arc(O1,2,0,60)),SW); label("b",midpoint(Arc(O2,1,85,180)),SE); label("c",midpoint(Arc(O1,2,60,180)),SE); label("d",midpoint(Arc(O2,1,0,85)),SW);[/asy]$

 1955 AHSC (Problems • Answer Key • Resources) Preceded byProblem 28 Followed byProblem 30 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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 All AHSME Problems and Solutions