MathDB

37

Part of 1954 AMC 12/AHSME

Problems(1)

Angles Bisected

Source:

2/7/2009
Given triangle PQR PQR with RS‾ \overline{RS} bisecting ∠R \angle R, PQ PQ extended to D D and ∠n \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("pp",P+(0.35,0.1)); draw(anglemark2(R,Q,P,11)); label("qq",Q+(-0.17,0.1)); draw(anglemark2(R,Np,D,8,true)); label("nn",Np+(+0.12,0.07)); draw(anglemark2(R,M,D,13,true)); label("mm",M+(+0.25,0.03)); draw(anglemark2(M,D,P,29)); label("dd",D+(-0.75,0.095));
pen f=fontsize(10pt); label("RR",R,N,f); label("PP",P,S,f); label("SS",Sp,S,f); label("QQ",Q,S,f); label("DD",D,S,f);[/asy] (A)\ \angle m \equal{} \frac {1}{2}(\angle p \minus{} \angle q) \qquad (B)\ \angle m \equal{} \frac {1}{2}(\angle p \plus{} \angle q) (C)\ \angle d \equal{} \frac {1}{2} (\angle q \plus{} \angle p) \qquad (D)\ \angle d \equal{} \frac {1}{2}\angle m \qquad (E)\ \text{none of these is correct}