Grades 9-10 Video Solutions 2013-Level 9&10 Video Solutions 2013 problem27

Video Transcription

Question number 27.  A regular 13-sided polygon is inscribed in a circle with center O. Triangles can be formed by choosing three vertices of this polygon to be the vertices of the triangle.  How many of the triangles that can be formed in this way contain the point O?  Here I have drawn a regular 13-sided polygon  and the center is indicated by the red dot and the vertices here are marked in green, lying on the circumscribing circle.  And so let's begin by noting that if we try to draw in any diameter of our circle, then we can at most have one vertex lying on the diameter.  And that will allow us to split the circle into two arcs, two semicircles, one containing six vertices and one containing seven vertices.  So let's make a note of that by drawing a diameter.  We have an arc with six vertices. So in our picture, that would be the one here to the left of the line and an arc with seven vertices.  And that's the one over here. Then what we will do is we will connect the farthest vertices on the arc with seven vertices together.  So these two will become one edge of our triangle.  And so let's make a note of that. There are seven ways to choose the edge connecting the farthest  vertices on this arc.  And to make sure that the triangle that I will finish drawing now contains the center, we choose the third vertex to lie somewhere  on the opposite arc, the one containing six vertices. So I'll just draw in something like this. And so here we have a triangle.  That we are looking for. Now how many ways can we choose the third vertex? It looks like we have six choices. But in fact, we have more. We can rotate.  So let's see if I can show that. I have a tool here that will allow me to make a rotation.  Let's try something like this.  And we see here that's one configuration of the triangle.  And I can rotate it a little bit. And then I have a second one.  And then third one, four, five, six, seven, and so forth. If I go all the way around, I have in fact 13 of them.  So there are 13 ways to choose the third vertex. And the total all together, we have 7 times 13 or 91  such triangles.  So the answer is C.

Video Summary

To find the number of triangles containing the center O of a regular 13-sided polygon inscribed in a circle, split the circle with a diameter creating arcs with six and seven vertices each. Select two farthest vertices from the arc with seven vertices to form one triangle side, with the third vertex from the arc with six vertices. There are 7 ways to select two farthest vertices and 13 positions for the third vertex due to rotational symmetry of the polygon, resulting in 7 x 13 = 91 such triangles. Thus, 91 triangles contain the center O.

Keywords

triangles

center O

13-sided polygon

circle

rotational symmetry