jmc

Question

ABCD is a rectangular sheet of paper that has been folded so that corner B is matched with point B' on edge AD. The crease is EF, where E is on AB and F is on CD. The dimensions AE=8, BE=17, and CF=3 are given. The perimeter of rectangle ABCD is m/n, where m and n are relatively prime positive integers. Find m+n. FIGURE

Accepted Answer

FIGURE Since EF is the perpendicular bisector of BB', it follows that BE = B'E (by SAS). By the Pythagorean Theorem, we have AB' = 15. Similarly, from BF = B'F, we havealign* BC^2 + CF^2 = B'D^2 + DF^2 & BC^2 + 9 = (BC - 15)^2 + 484 BC &= 703 align*Thus the perimeter of ABCD is 2(25 + 703) = 2903, and the answer is m+n=293.