Though I'm not particularly fast, I would still call myself a runner. Many non-runners wonder how runners can possibly entertain themselves over the course of a 5-mile run. To be honest, I don't normally remember what goes through my head, but today it was trapezoids.

Specifically, given trapezoid ABCD (where AB || CD), how can you prove that CA + AB + BD is greater than CD?

It seems obvious that the sum of the other three sides should be longer (even if CA and BD are really short), but that doesn't mean it doesn't merit a proof! Drawing a single diagonal lends itself to a simple proof using the triangle inequality:

This yields the following two inequalities:

(1) CA + AB > BC

(2) BC + BD > CD

If we add BD to both sides of (1) we have:

(3) CA + AB + BD > BC + BD

Combining with (3) we have:

(4) CA + AB + BD > BC + BD > CD

So this proves that CA + AB + BD > CD!

I was thinking about this while running around the Charles River this morning, wondering whether it would ever be shorter to run past a long bridge (CD) to cross the river at a shorter bridge (AB) and then come back, but now we've proved that is never the case!

I only used a trapezoid because that more closely matched the geometry of what I was running, but I think it's trivial to extend the proof to any quadrilateral (though ones that are not convex may not cooperate).

## Wednesday, October 21, 2009

Subscribe to:
Post Comments (Atom)

This is one of the many reasons I'm happy to have you as my friend. Thanks for the smile.

ReplyDeleteWow. Just. Wow.

ReplyDeletethe proof works for any polynomial

ReplyDeleteHere's another proof.

ReplyDelete1. On a Euclidean surface, the shortest distance between two points is a straight line.

2. Therefore, CD is the shortest distance between C and D.

3. CA + AB + BD is not a straight line between C and D.

4. Therefore, CA to AB to BD is not the shortest distance between C and D.

5. Therefore, CD is shorter than AB + BC + CD.

Finding the best online history research paper writing services and History Research Paper Services is not easy unless one is keen to establish a professional history assignment writing service provider & history coursework help online.

ReplyDeletePsychology coursework writing services are not hard to come across for those in need of Psychology Research Paper Services and psychology assignment writing services.

ReplyDeleteThank you for all of your work on this web page.

ReplyDeleteUFAABET

Numerous non-sprinters can't help thinking about how sprinters can engage themselves throughout the White Board Animation Services span of a 5-mile run. Truth be told, I don't typically recollect what experiences my head, yet today it was trapezoids. I just utilized a trapezoid since that all the more firmly coordinated the calculation of what I was running.

ReplyDeleteThanks for sharing a good content like this.It’s very informative for everyone.룰렛

ReplyDeleteAmazing and infromative post, checkout this as well:- online 8th grade math tutor

ReplyDeletebrillant piece of information, I had come to know about your web-page from my friend hardkik, chennai,i have read atleast 9 posts of yours by now, and let me tell you, your webpage gives the best and the most interesting information. This is just the kind of information that i had been looking for, i'm already your rss reader now and i would regularly watch out for the new posts, once again hats off to you! Thanx a million once again

ReplyDelete토토

프로토

토토사이트

these websites are really needed, you can learn a lot.

ReplyDelete토토사이트

스포츠중계

토토

The assignment submission period was over and I was nervous, casino online and I am very happy to see your post just in time and it was a great help. Thank you ! Leave your blog address below. Please visit me anytime.

ReplyDeleteThank you for giving me this amazing information. You can also find more here 안전놀이터추천

ReplyDelete