The Student Will investigate and prove the triangle inequality

The Student Will determine when triples of numbers can make triangles

Prior
Knowledge: Inequalities in one triangle; properties of inequalities

Motivational
Question: An isosceles triangle has sides of length 10, 10, and 26. What is the area of the triangle?

Overview: Hopefully the students will mindlessly drop the perpendicular from the vertex angle and then try to solve the resulting right triangle for the height. Of course, this triangle can not exist--the anomalous results will hopefully lead to a constructive discussion about what is required of the side lengths to form a triangle.

Typical proofs of the triangle inequality involve dropping an altitude to the longest side (guaranteed to be contained inside the triangle) and then using the fact that the perpendicular is the shortest distance from a point to a line.

Use the triangle inequality to determine if a given triple of numbers could make a triangle; then, use abstract expressions (like x, x+1, x+2 etc). Using both directions of the triangle inequality, one can determine the minimum and maximum of a missing side.

Interesting extension questions might involve applying the triangle inequality to quadrilaterals by constructing diagonals. Or given two sides of the triangle, how many triangles with all integer sides can be created? Even more interesting is to investigate what happens to known formulas (like Heron's Formula) when illegitimate side lengths (i.e. that do not satisfy the triangle inequality) are given.

A fun and quick activity is to pass out one notecard to every student, each with a number or expression on it, and give the students 30 seconds to find two partners with whom they form a legitimate triangle. Then repeat.

Key
Questions: What is wrong in our motivational problem? What does the triangle inequality say about the sides of a triangle? What does the triangle inequality not say about the sides of a triangle? Is there a quadrilateral inequality?

Looking Ahead: The triangle inequality is a crucial check to determine if a solved triangle is legitimate (say in a trigonometry problem). The triangle inequality is also fundamental in the geometry of vectors.