1971 AHSME Problems/Problem 5

Problem 5

Points $A,B,Q,D$ , and $C$ lie on the circle shown and the measures of arcs $\widehat{BQ}$ and $\widehat{QD}$ are $42^\circ$ and $38^\circ$ respectively. The sum of the measures of angles $P$ and $Q$ is

$\textbf{(A) }80^\circ\qquad \textbf{(B) }62^\circ\qquad \textbf{(C) }40^\circ\qquad \textbf{(D) }46^\circ\qquad \textbf{(E) }\text{None of these}$

$[asy] size(3inch); draw(Circle((1,0),1)); pair A, B, C, D, P, Q; P = (-2,0); B=(sqrt(2)/2+1,sqrt(2)/2); D=(sqrt(2)/2+1,-sqrt(2)/2); Q = (2,0); A = intersectionpoints(Circle((1,0),1),B--P)[1]; C = intersectionpoints(Circle((1,0),1),D--P)[0]; draw(B--P--D); draw(A--Q--C); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SW); label("$D$",D,SE); label("$P$",P,W); label("$Q$",Q,E); //Credit to chezbgone2 for the diagram[/asy]$

Solution 1

We see that the measure of $P$ equals $(\widehat{BD}-\widehat{AC})/2$ , and that the measure of $Q$ equals $\widehat{AC}/2$ . Since $\widehat{BD} = \widehat{BQ} + \widehat{QD} = 42^{\circ} + 38^{\circ} = 80^{\circ}$ , the sum of the measures of $P$ and $Q$ is $\widehat{BD}/2 = 80^{\circ}/2 = 40^{\circ} \Longrightarrow \boxed{\textbf{(C) }40^{\circ}}$ .

Solution 2

Arcs are measured by the angle measures of their corresponding central angles. Thus, the inscribed angle $\measuredangle BAQ = \tfrac{42^{\circ}}2 = 21^{\circ}$ , and, likewise, $\measuredangle QCD = \tfrac{38^{\circ}}2 = 19^{\circ}$ . Thus, by supplementary angles, $\measuredangle PAQ = 180^{\circ} - \measuredangle BAQ = 159^{\circ}$ , and $\measuredangle PCQ = 180^{\circ} - \measuredangle QCD = 161^{\circ}$ . Because the sum of the interior angle measures of a quadrilateral add to $360^{\circ}$ , we see that $\measuredangle P + \measuredangle Q = 360^{\circ} - \measuredangle PAQ - \measuredangle PCQ = 360^{\circ} - 159^{\circ} - 161^{\circ} = 40^{\circ}$ . Thus, our answer is $\boxed{\textbf{(C) }40^{\circ}}$ .

1971 AHSC (Problems • Answer Key • Resources)
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1971 AHSME Problems/Problem 5

Contents

Problem 5

Solution 1

Solution 2

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