Puzzle: McRib
2007 November 18
McRib? Is that like McNuggets? Nope. ‘McRib‘ is my friend. In fact, years ago, he had a thing for 2 for a dollar McRibs, but we all eat better today. Today. Today McRib has a broken rib, and, while stuck flat on his back, considered the following question:
Given a circle inscribed in a right trapezoid, express the radius of the circle in terms of the bases of the trapezoid.
Inscribed = four points of tangency. Right trapezoid = quadrilateral, exactly 2 parallel sides (bases), one of the other sides perpendicular to the bases.
Put discussion and questions in the comments section, below.
Please put answers (and discussion of answers) here (in a separate post). The answer is ’surprisingly simple’ and I would like to learn why.
Unless I’m missing something in the setup here, I don’t think you can express the radius in terms of the bases.
Explanation: Simplest case is a square with a circle inscribed. Radius is a quarter of the sum of the bases. That is, r = (b1 + b2) / 4. Start from that picture.
Without changing the height of the trapezoid or radius of the circle, change the angle between a base and the non-perpendicular side.
One base gets smaller and the other gets larger. Now imagine what happens as the point of intersection between a base and this angled side approaches the point of tangency between that base and the circle. The length of that base approaches r… and the other base goes to infinity. But r hasn’t changed.
Am I thinking of something different than you’re thinking of?
Huh, threw this into Sketchpad (which is much more fun than what I was doing with it). Having trouble finding a relationship for reasons Ben described above. What am I missing?
But… a circle is determined by three points, so in general you can’t necessarily inscribe a circle in any old right trapezoid. And for those that you can, the diameter of the circle is necessarily the same as the distance between the two parallel sides, so it doesn’t have anything to do with the length of the bases.
Nov 19, 3:19 AM — [ Edit | Delete | Unapprove | Approve | Spam ] —
Ben, Jackie, Brent…
it can’t be “any old right trapezoid” for the reasons you mention. So, of the right trapezoids that are tangent to a circle at 4 points, express the radius in terms of the bases.
Yeah, but there are an infinite number of right trapezoids for which this holds. Hold on, let me find a place to link the sketchpad file.
Yes, there are. But there is a fixed relationship between the bases and the radius, which is (1) surprising, and (2) surprisingly simple. David Radcliffe posted that relationship (don’t look at the solution post, if you haven’t seen it yet and still want to look for it).
Or, go look at David’s (correct) solution, and help me. I want to know about the “why” of the question.
Well, I don’t know why I felt the need to leave the comment before uploading the file. Anyway, here is the java sketchpad file.
I still don’t see what I’m missing.
Now I am really frustrated. That file is great, and allows me to look at lots of numerical verification of the solution, which is pleasing, but it doesn’t help me understand why this is happening.
I think I need to grade, then think about it tomorrow.
Oh no whys (yet). I just wanted to make sure we were talking about the same thing.
jonathan,
Great problem! I posted some comments, a brief solution and connections to a related problem in your solutions area. Happy Thanksgiving!
Dave