Using problem solving skills, formal geometry, and The Geometer’s Sketchpad, students will model and solve a real-world problem by constructing the centroid of a triangle. Students will explore further relationships of the medians of a triangle and investigate how the centroid partitions each of the medians.  

Mathematics: Students will investigate the medians of a triangle and the conjecture the following theorems:

• A median divides a triangle into two triangles with equal area.

• The three medians of any triangle are concurrent.

• The three medians of a triangle divide the triangle into six triangles with equal area.

• The centroid in a triangle divides each median into two parts, whose lengths are in a 2:1 ratio.

Extension activities include investigating the Euler line for right, isosceles and equilateral triangles.

Mathematical Thinking: Students will be asked to predict, conjecture and prove the aforementioned theorems.

Technology: This activity uses several Sketchpad commands such as: midpoint of a segment, polygon interior, point at intersection, area, perimeter and distance.

The following sketch illustrates the use of the centroid to divide a triangle into three equal areas.