1954 AHSME Problems/Problem 37

Problem 37

Given $\triangle PQR$ with $\overline{RS}$ bisecting $\angle R$ , $PQ$ extended to $D$ and $\angle n$ a right angle, then:

$[asy] path anglemark2(pair A, pair B, pair C, real t=8, bool flip=false) { pair M,N; path mark; M=t*0.03*unit(A-B)+B; N=t*0.03*unit(C-B)+B; if(flip) mark=Arc(B,t*0.03,degrees(C-B)-360,degrees(A-B)); else mark=Arc(B,t*0.03,degrees(A-B),degrees(C-B)); return mark; } unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair P=(0,0), R=(3,2), Q=(4,0); pair S0=bisectorpoint(P,R,Q); pair Sp=extension(P,Q,S0,R); pair D0=bisectorpoint(R,Sp), Np=midpoint(R--Sp); pair D=extension(Np,D0,P,Q), M=extension(Np,D0,P,R); draw(P--R--Q); draw(R--Sp); draw(P--D--M); draw(anglemark2(Sp,P,R,17)); label("$p$",P+(0.35,0.1)); draw(anglemark2(R,Q,P,11)); label("$q$",Q+(-0.17,0.1)); draw(anglemark2(R,Np,D,8,true)); label("$n$",Np+(+0.12,0.07)); draw(anglemark2(R,M,D,13,true)); label("$m$",M+(+0.25,0.03)); draw(anglemark2(M,D,P,29)); label("$d$",D+(-0.75,0.095)); pen f=fontsize(10pt); label("$R$",R,N,f); label("$P$",P,S,f); label("$S$",Sp,S,f); label("$Q$",Q,S,f); label("$D$",D,S,f);[/asy]$

$\textbf{(A)}\ \angle m = \frac {1}{2}(\angle p - \angle q) \qquad \textbf{(B)}\ \angle m = \frac {1}{2}(\angle p + \angle q)$ $\textbf{(C)}\ \angle d =\frac{1}{2}(\angle q+\angle p)\qquad \textbf{(D)}\ \angle d =\frac{1}{2}\angle m\qquad \textbf{(E)}\ \text{none of these is correct}$

Partial Solution

path anglemark2(pair A, pair B, pair C, real t=8, bool flip=false)
{
 pair M,N;
 path mark;
 M=t*0.03*unit(A-B)+B;
 N=t*0.03*unit(C-B)+B;
 if(flip)
 mark=Arc(B,t*0.03,degrees(C-B)-360,degrees(A-B));
 else
 mark=Arc(B,t*0.03,degrees(A-B),degrees(C-B));
 return mark;
}
unitsize(1.5cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
pair P=(0,0), R=(3,2), Q=(4,0);
pair S0=bisectorpoint(P,R,Q);
pair Sp=extension(P,Q,S0,R);
pair D0=bisectorpoint(R,Sp), Np=midpoint(R--Sp);
pair D=extension(Np,D0,P,Q), M=extension(Np,D0,P,R);
draw(P--R--Q);
draw(R--Sp);
draw(P--D--M);
pair pI=extension(D,M,R,Q);
void label("$O$",pI,f);
draw(anglemark2(Sp,P,R,17));
label("$p$",P+(0.35,0.1));
draw(anglemark2(R,Q,P,11));
label("$q$",Q+(-0.17,0.1));
draw(anglemark2(R,Np,D,8,true));
label("$n$",Np+(+0.12,0.07));
draw(anglemark2(R,M,D,13,true));
label("$m$",M+(+0.25,0.03));
draw(anglemark2(M,D,P,29));
label("$d$",D+(-0.75,0.095));
pen f=fontsize(10pt);
label("$R$",R,N,f);
label("$M$",M+(-.067,.067),f);
label("$N$",Np+(-.07,.14),f);
label("$P$",P,S,f);
label("$S$",Sp,S,f);
label("$Q$",Q,S,f);
label("$D$",D,S,f); (Error making remote request. Unknown error_msg)

Looking at triangle PRQ, we have $\angle RPD+\angle RQS+\angle MRN=180$ and from the given statement $\angle NMR=\frac{1}{2}\angle MRN$ , so looking at triangle $\angle NMR=90-\frac{}{}$

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1954 AHSME Problems/Problem 37

Problem 37

Partial Solution