Grades 11-12 Video Solutions 2010-11&12 Video Solutions 2010 problem17

11&12 Video Solutions 2010 problem17

Video Transcription

Video Summary

The problem involves finding the number of ways to choose three vertices of a regular 14-gon to form a right triangle. By inscribing the 14-gon in a circle and using Thales' theorem, we know a right triangle is formed if its hypotenuse is a diameter of the circle. To find the right triangles, we calculate the number of distinct diameters, which is the combination of 14 vertices taken two at a time, adjusted for symmetry. Solving \( \binom{14}{2} - 7 \) gives us the number of unique right triangles: 84. Thus, 84 such triangles can be formed.

Keywords

right triangle

14-gon

Thales' theorem

diameter

combinatorics