| An 8 x 8 square is divided into four pieces, two trapezoids and two right triangles. The area of the square is 64. |  |
Now, rearrange the four pieces creating a 5 x 13 rectangle, a rectangle with an area of 65. |
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| Can this really happen? Please explain. | |
Solution by Nicolas Pueppke , Williamston High School
The pieces A , B , C , and D cannot be rearranged as shown to make up a 
rectangle with an area of 65.
In order for this rectangle to exist, both the triangle formed by A and C and the triangle formed by B and D must be congruent right triangles. However, this is impossible. In fact, the "triangles" are not triangles at all, but instead they are four-sided figures. This can easily be proven by computing the tangent because in order for the larger triangles to work, angles x and y must be congruent. In order for the angles to be congruent, their tangents must be equal. Now, 
and 
, and 
.
Also solved by: Eric Anderson, Lauren Bacans, Clare Boals, Brooke Boer, Brad Brennan, Allison Bryan, Max Dixon, Mischa Dylewski, Blaire Edgerton, Jon Foster, Brianna Kearney, Ryan Klock, Kristin Kopec, Adam Leenheer, Amanda Ly, Christy Maier, Hailey Novak, Corinne O'Grady, Katie Olson, Victoria Raymond, Andrew Rennaker, Drew Reylelts, Aaron Sayfie, Nick Sayles, Caroline Sierra, Sam Swardenski, Dani Tyner, Jill Trierweiler, Gabrielle Whip, Sarah Wolfson, Alex Zheutlin, Philip Zimbardo (Forest Hills Eastern H.S.) ; Mary Jo Jeffers, Andrew Koo, Emily Ly, Marienne Mbaye, Alex Mead, Tessa Morris, Elizabeth Potter, Brian Siebers, Medina Skokic, Barret Vollmer, Maurice Wong (Forest Hills Northern H.S.) ; Benjamin Blanchard, Morgan Brown, Kaitlyn Brock, Alissa Coldwell, James Custer, Justin Travis, Kelsey Travis, Sheila Wheeler, Hannah Wilcox (Shepherd H.S.) ; Josh Buscher, Shannon Liu, Josh Lovatt, Anshumon Sarungi, Kyle Williams (Novi H.S.) ; Andrew Clark, Brandon Lothschutz, Josh Schumaker (Coopersville H.S.) ; Robert Harrison, Alex Robinson (Lansing Catholic H.S.) ; Nick Wasylyshyn (Canton H.S.) ; Garrett Rockafellow (Central Montcalm H.S.) ; Phil Dirkse (Jenison H.S.) ; David Maniez (St. Louis H.S.) ; Steve Draggoo (East Lansing H. S.) ; Armila Francis (Plymouth Christian Academy) ; Chris Struble (Alamo Heights H.S., San Antonio, TX) ; and Merideth Voree (no school given).