MathDB

ABCD

is a rectangle (see the accompanying diagram) with

P

any point on

\overline{AB}

\overline{PS} \perp \overline{BD}

and

\overline{PR} \perp \overline{AC}

\overline{AF} \perp \overline{BD}

and

\overline{PQ} \perp \overline{AF}

. Then PR \plus{} PS is equal to: [asy]defaultpen(linewidth(.8pt)); unitsize(3cm);
pair D = origin; pair C = (2,0); pair B = (2,1); pair A = (0,1); pair P = waypoint(B--A,0.2); pair S = foot(P,D,B); pair R = foot(P,A,C); pair F = foot(A,D,B); pair Q = foot(P,A,F); pair T = intersectionpoint(P--Q,A--C); pair X = intersectionpoint(A--C,B--D);
draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(P--S); draw(A--F); draw(P--R); draw(P--Q);
label("

A

",A,NW); label("

B

",B,NE); label("

C

",C,SE); label("

D

",D,SW); label("

P

",P,N); label("

S

",S,SE); label("

T

",T,N); label("

E

",X,SW+SE); label("

R

",R,SW); label("

F

",F,SE); label("

Q

",Q,SW);[/asy] (A)\ PQ\qquad (B)\ AE\qquad (C)\ PT \plus{} AT\qquad (D)\ AF\qquad (E)\ EF

47

Problems(1)