Grades 9-10 Video Solutions 2010-Levels 9&10 Video Solutions 2010 problem25

Video Transcription

Question number 25. Point A0 lies on one of the rays, which form an angle of 9 degrees with the  vertex here O. Starting at the point A0, we draw segments of length equal to the distance between  O and A0, connecting points different from O on alternating sides of the angle as shown. So what  we have is the same segment, same distance, is being repeated here in green once, twice, a third  time, like that. So let me draw in the missing edges here. And where these come together on the  angle, we have a point with an increasing index. And what we are creating are triangles. And I  will just call these triangles by the point A sub, a number, lying on the rays. So triangles  A0, A1, A2 are isosceles by construction. And we can keep track of the angles. So A0 has two  9-degree angles, the one pictured, and then the congruent one here. And the central angle here  at A0 has to measure 162 degrees. That's 180 minus 9 minus 9. And we can continue. A1 will have here  an angle that is the supplement of 162 degrees. So an 18-degree angle here, an 18-degree angle  there. So we have 18, 18, and then 180 minus 36 is 144 degrees. That angle is at A1. And then we can go  on and compute the angle here in orange. That would be 180, we subtract 144, and also we subtract 9.  And that gives us an angle of 27 degrees. So A2 has two 27-degree angles, and then the central angle  at A2 measures 126 degrees, like so. And then we can keep going, noticing that the measure of the  two equal angles here is a multiple of 9. And so, in general, what we have is that A sub n here would  have two angles measuring 9, and we need to multiply 9 by n plus 1. So that's that number of degrees. So the  largest angle that we can create here is when n is equal to 9. So A9 would have two 90-degree angles, and then  the third angle does not exist. So in fact, this is not a triangle. So here the process stops. But the step before  that, we have two 81-degree angles, and then the last angle is possible. It measures 18 degrees. So from 0 to 8, we have a  list here of valid triangles, and that is nine valid triangles. And that answers the questions. At most, how many  segments can be drawn? And we can draw nine, thereby creating nine triangles like that, with angle measures that add up to 180 degrees and not more. So the answer here is 9b.

Video Summary

In this problem, segments are drawn from a point A0 on one ray of a 9-degree angle, creating isosceles triangles with alternating sides. Each step increases the index of the triangle, forming angles at each point. Starting from two 9-degree angles at A0, the central angle is calculated by subtracting the known angles from 180 degrees. The process continues with each new point, where the two identical angles in each triangle are multiples of 9. The maximum number of triangles formed, before one angle becomes non-existent, is nine. Therefore, nine segments can be drawn, forming nine valid triangles.

Keywords

isosceles triangles

9-degree angle

central angle

triangles formation

maximum segments