Grades 11-12 Video Solutions 2013-Grades 11-12 Video Solutions 2013 problem2

Video Transcription

Question number two.  The large regular octagon in the figure measures 10 on each side, which is the measure of the radius of the circle inscribed in the smallest octagon formed by the diagonals.  To begin, let's mark the distance 10 on the diagram.  That's the length of the side here, and it appears that that is also the diameter of the inscribed circle.  Now, since the circle is inscribed,  the diagonals here are tangent to it. So the red lines I'm marking are tangent to the circle.  And to conclude that 10 is the diameter of the inscribed circle, we must check that the red lines are in fact parallel.  And so we draw in a diameter of the circle circumscribing the large octagon.  So that would be a line passing through the center of the inscribed circle and connecting opposite vertices of the large octagon like so.  We can then complete that to a triangle.  And note that by an old theorem in geometry sometimes attributed to Thales. The angle marked here in the corner is a right angle.  And so it must be the one that's in the opposite corner by the same result.  And we're looking now at a rectangle. So indeed, the red lines are parallel. The distance between them is 10. That's the diameter of the inscribed circle.  And so its radius must be 5.

Video Summary

The problem involves finding the radius of a circle inscribed within the smallest octagon formed by diagonals inside a larger regular octagon, each side measuring 10. The speaker establishes that the circle's diameter equals the octagon's side length, 10, by showing the red lines, which are tangent to the circle, are parallel. Using geometric theorems, they form a rectangle to confirm that these lines are indeed parallel. Consequently, if the diameter of the inscribed circle is 10, the radius is 5.

Keywords

inscribed circle

octagon

geometry

radius

diagonals