Activity Overview
Students produce three relatively simple proofs relating to randomly generated chords in a cirlce. The problem is that the three proofs produce three different results. Can you solve this chord conundrum?
Objectives
Students arrive at a conundrum, known as Betrand's Problem. How is it possible to prove three different probabilities for the same problem. The purpose of the activity is to encourage students to look deeper into definitions. The brilliant video by Grant Sanderson (Three Blue - One Brown) on this topic provides a great companion to this activity.
Vocabulary
- Geometric proof
- Computer simulation
- Contradiction - Conundrum
About the Lesson
A chord is randomly generated in a circle. What would be the average length of the chord? To simplify this problem, the question is changed slightly to compare the length of the chord to an equilateral triangle inscribed in the circle. In this activity three different ways are used to generate the chord, the problem is, they arrive at three different solutions to the problem! Students explore the geometric proofs to the computer generated results. Can you solve the conundrum?