Grades 11-12 Video Solutions 2021-video 2021 11-12/26

video 2021 11-12/26

Video Transcription

On a circle, 15 points are equally spaced. We can form triangles by joining any three of these.  We count two triangles as being the same if they are congruent, i.e. one is a rotation and or a reflection of the other. How many different triangles can be drawn?  So we have 15 evenly spaced points on the circle that we could have a vertex of a triangle on. And we can choose three unique ones to define a triangle.  So how many different triangles can we make? Well, we can use combinations to say that we have 15 points and we want to choose three.  The formula for this is 15 factorial over 3 factorial times 15 minus 3 factorial,  but we don't actually need to compute 15 factorial since a lot of the terms will end up canceling out. Here, a lot of the terms in 15 factorial and 12 factorial will cancel out.  So we will just be left with 15 times 14 times 13 over 3 factorial, which is still a little much but can be computed. If you end up multiplying it out, you get 445,  which is the number of possible triangles that could be made. Now we need to take out all the ones that are congruent. So first let's start out with equilateral triangles.  Five of these triangles could be equilaterals. You can make these by choosing three evenly spaced out points, and you have five options.  However, these would only result in one unique triangle since all the other triangles would just be rotated versions of that equilateral triangle.  Next, let's discuss isosceles triangles. We have six variants of isosceles triangle, not including the equilateral one,  and each has 15 possible rotations or 16 possible starting points. So we have 6 times 15 possible isosceles combinations,  but only six of them would be unique as the remaining ones would be the same isosceles triangles, just rotated.  So of our original 455 triangles, we have found 5 equilaterals and 90 isosceles triangles. This leaves us with 360 scaling triangles.  Each of these triangles could be rotated 15 different ways, and each of them could have two reflections, either no reflection or a reflection across the diameter.  So each scaling triangle has 15 times 2 congruent variants that are just rotated reflections or both.  So 30 congruent variants. So if we have 360 possible scaling triangles and 30 combinations for each, that means that we only have 12 unique scaling triangles.  So we have 12 unique scaling triangles, 6 unique isosceles triangles, and 1 unique equilateral triangle. So that gets us a total of 19 unique triangles.  So the question asked us, how many different triangles can be drawn? The answer is 19. Letter A.

Video Summary

The problem involves determining the number of unique triangles that can be formed by joining any three of 15 equally spaced points on a circle, where congruent triangles (by rotation or reflection) are considered the same. There are typically 455 ways to choose 3 points from 15 (using combinations), but many of these are congruent. By analyzing the shapes—5 equilateral, 6 unique isosceles, and 12 unique scalene triangles—and accounting for various congruences, the solution reveals there are 19 unique triangles that can be formed. Thus, the answer is 19 different triangles.