Rhombus Minimization
Source:
January 5, 2009
geometryrhombusinequalitiessimilar trianglesPythagorean Theorem
Problem Statement
Let be a rhombus with AC\equal{}16 and BD\equal{}30. Let be a point on , and let and be the feet of the perpendiculars from to and , respectively. Which of the following is closest to the minimum possible value of ?
[asy]unitsize(2.5cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));pair D=(0,0), C=dir(0), A=dir(aSin(240/289)), B=shift(A)*C;
pair Np=waypoint(B--A,0.6), P=foot(Np,A,C), Q=foot(Np,B,D);draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
draw(Np--Q);
draw(Np--P);label("",D,SW);
label("",C,SE);
label("",B,NE);
label("",A,NW);
label("",Np,N);
label("",P,SW);
label("",Q,SSE);draw(rightanglemark(Np,P,C,2));
draw(rightanglemark(Np,Q,D,2));[/asy]