Chapter 5

Geometric Inequalities

Two paths lead to the same gate, and you want to know which is shorter. A sketch might suggest an answer, but what if it is not drawn to scale and no distances are labeled? Geometric inequalities let us ask what a shape can tell us about sizes when exact measurements are missing. A comparison may be within reach even when a calculation is not. What would make that comparison convincing?

What You'll Explore

3 lessons.

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1. 1
   Sides, Angles, and Shortest Distances →

   Done Skipped Revisit Clear ▾

   Can a triangle tell you which of its sides is longer without giving you their lengths? Explore connections between sides and angles, then ask how those connections can help you compare different distances from a point to a line.

2. 2
   The Triangle Inequality →

   Done Skipped Revisit Clear ▾

   Three sticks are ready to form a triangle, but can their ends always meet? Explore the limits on possible side lengths and investigate what the same puzzle can tell you about routes between two points.

3. 3
   The Hinge Theorem →

   Done Skipped Revisit Clear ▾

   Picture two triangles with matching pairs of side lengths, but different angles between those sides. What might comparing those angles tell you about the remaining sides, and could comparing the sides tell you something about the angles?