A triangle with sides of lengths 10, 17, and 21 has a square inscribed in it. One side of the square lies on the longest side of the triangle. The other two vertices of the square touch the two shorter sides of the triangle. What is the length of the side of the square? Hint: Heron's formula!
Solution by Jerrod Kempf , Charlotte High School
Using Heron's formula we can find the area 
square units. Assuming the side of length 21 is the base, this would mean the triangle's height is 8. Now we need to find a formula for the area of the figure that involves s , the side length of the square. To do this, divide the figure into 2 pieces, the top triangle and the rest of the figure, which forms a trapezoid. The top triangle has a base of s and a height of 
; so its area is 
. The area of the trapezoid is: 
.
The problem can now solved as: 
Also solved by: Evan Laske (Bishop Foley Catholic H.S.); Aaron Lange (Bullock Creek H.S.); Trevor Barker, Yuan Chenglin, Raoul Marx, Matt Yenneman (Central Montcalm H.S.); Galina Gagin (East Grand Rapids H.S.); Ananya Avinash, Sarah Caponi, Ari Presant, Walter Tan, Nico Wagner (Forest Hills Central H.S.); Meredith Assaly, Lisa Bloom, Brooke Boer, Garrett Cullen, Shelby Eagle, Melvin Joice, Ellen Junewick, Sami Khodor, Miles Korth, Ryan Klock, Kristin Rybski, Kailene Schabes, Morgan Smith, Gina Veltman, Alex Zheutlin (Forest Hills Eastern H.S.); Sarah Bakale, Caroline Bjerregaard, Murali Tej Bolla, Jack Duiven, Tyler Kamm (Forest Hills Northern H.S.); Adriana Combs, Dayna Diskin, Taylor Fisher, Britt Gabel, Rachel Holloway, Caitlin McCarthy, Ashley Morgan, Sean Rilett, Nate Weller (Linden H.S.); Robert Harrison, Alex Robinson (Lansing Catholic H.S.); Ashley Meyer (Shepherd H.S.); Megan Greve, Paul Kitti (Plymouth Christian Academy); Sira Baek, Niharika Goparaju, Allison Liao, Rie Tomida (Troy H.S.); David Maniez (St. Louis H.S.); Kentaro Nishizaki (West Bloomfield H.S.); Kristen Katje, Jennifer Melorhe, John Schuitema, Matt Vermalre, Kari Witte [Western Michigan Christian H.S.]; Bruce Kraay (Zeeland East H.S.); Louise Avellar, Gio Campagna, Johnny Gragg, Louie Karam, Kristy Laugen, Alex Shaffer, Nicholas Shaffer, Mark Stephens, August Sullivan (Alamo Heights H.S., San Antonio, TX)