Published on August 24, 2011

#### Objectives

- Calculate the determinant of a matrix
- Evaluate a determinant to calculate the area of a triangle or quadrilateral in the plane, given the coordinates of its vertices

#### About the Lesson

The Case of the Missing Square presented here, is one of the classic paradoxes in recreational mathematics. The recent Sherlock Holmes movie provides a “modern” context for solving this paradox as a criminal case investigation. The assumption that the line joining the lower left corner of the rectangle to the upper right corner is a straight line results in a paradox. The sum of the areas of the four regions into which the rectangle is divided appears to be 64 square units, while the area of the original rectangle is 65 square units. The paradox is resolved when it is discovered that this “diagonal” line is not a straight line, but three line segments. Therefore the regions Gamma and Delta are not triangles, and all four regions are quadrilaterals. This sets the stage for the introduction of determinants to calculate the areas of the quadrilaterals given the coordinates of their vertices.

© Brendan Kelly Publishing, Inc. Reprinted with permission. www.brendankellypublishing.com