MathDB
Paper Folding

Source:

November 17, 2005
geometryperimeterrectanglenumber theoryrelatively primeperpendicular bisector

Problem Statement

ABCDABCD is a rectangular sheet of paper that has been folded so that corner BB is matched with point B′B' on edge ADAD. The crease is EFEF, where EE is on ABAB and FFis on CDCD. The dimensions AE=8AE=8, BE=17BE=17, and CF=3CF=3 are given. The perimeter of rectangle ABCDABCD is m/nm/n, where mm and nn are relatively prime positive integers. Find m+nm+n. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C; draw(F--D--A--E); draw(E--B--C--F, linetype("4 4")); filldraw(E--F--Cp--Bp--cycle, white, black); pair point=( 12.5, 35/3 ); label("AA", A, dir(point--A)); label("BB", B, dir(point--B)); label("CC", C, dir(point--C)); label("DD", D, dir(point--D)); label("EE", E, dir(point--E)); label("FF", F, dir(point--F)); label("B′B^\prime", Bp, dir(point--Bp)); label("C′C^\prime", Cp, dir(point--Cp));[/asy]
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