Grades 11-12 Video Solutions 2024-2024_11-12

Video Transcription

12 points are equally spaced on a circle. How many triangles containing a 45 degree angle can be formed by choosing three of these points?  We know that the subtended angle from two points on a circle is half the angle created by those two points. Then, in this circle, a 45 degree angle can be created by any two  points that are 90 degrees apart. Let these two points be A and B and let the third be C, as shown in our figure below.  For any of the 12 choices of A and B, there are 8 choices for C. Thus, we have 12 minus 8 equals 96 possible triangles. However, we have double counted the 12 isosceles triangles  and then the number of distinct triangles we have is 96 minus 12, which is 84.

Video Summary

Twelve points are evenly spaced on a circle. We aim to determine how many triangles containing a 45-degree angle can be formed. A 45-degree angle can be formed by two points that are 90 degrees apart. For each pair of such points, there are 8 options for the third point, yielding 12 * 8 = 96 potential triangles. However, isosceles triangles, formed when the third point coincides with one of the initial points creating the 45-degree angle, have been counted twice. Subtracting these 12 duplicate isosceles triangles leaves us with a total of 84 distinct triangles.

Keywords

circle

triangles

45-degree angle

isosceles

geometry