Ponder This Challenge - February 2005 - Infinite right triangles

Ponder This Challenge:

Puzzle for February 2005

Here's a geometric puzzle. All three parts need to be solved.

Start with a right triangle T1 whose vertices are (0,0), (0,1), (1,1). The angle at (0,0) is /4. Use the hypotenuse of T1 as the leg of a new triangle T2, whose angle at (0,0) is /8, and which extends down towards the x-axis. Use the hypotenuse of T2 as the leg of a new triangle T3, whose angle at (0,0) is /16. Continue ad infinitum.

The angles at (0,0) form an infinite series which sums to /2, so the limit of the process is a line segment on the x-axis.

Part 1: How long is that segment?

Part 2: Find an equation in polar coordinates (r,) of a curve passing through all the vertices of the resulting triangles, except possibly (0,0).

Part 3: Find an equation for that curve in rectangular coordinates (x,y).

The first 100 people who answer correctly will be listed. The answer will be posted a week after the 100th is received, or at the end of the month.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solution

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  Answer:

  Let r[k] denote the length of the hypotenuse of triangle Tk, so that r[k-1] is the length of one leg of Tk, and r[0]=1 for starters. We have r[k-1]/r[k]=cos((/2)/2^k). This recurrence relation, along with initial condition r[0]=1, is solved by r[k]=(2^k)sin((/2)/2^k). (Apply the formula for sin(2b)=2sin(b)cos(b) to b=((/2)/2^k).) Restate this as r[k]=(/2)(sin(c[k])/c[k]) where c[k]=(/2)/2^k, notice that c[k] tends to 0 as k increases, and use limit(sin(z)/z)=1 as z tends to 0, to deduce that r[k] tends to the limit p/2, which answers Part 1.

  Notice that in the above expression, c[k] is also the angle between the x-axis and the hypotenuse of triangle Tk. So the point with polar coordinates (r,)=(r[k],c[k]) is one of the vertices of triangle Tk. A curve passing through all such vertices is given by r=(/2)(sin()/). This answers Part 2.

  Converting this to rectangular coordinates, we find y/x = tan [ y / 2(x^2+y^2) ] , to answer Part 3.

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  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• Tomas G. Rokicki (01.31.2005 @01:36:38 PM EDT)
• Michael Nedzelsky (01.31.2005 @05:01:19 PM EDT)
• Max Alekseyev (01.31.2005 @06:45:03 PM EDT)
• Tamas (01.31.2005 @07:06:33 PM EDT)
• Visegrady (01.31.2005 @07:06:33 PM EDT)
• Eugene Vasilchenko (01.31.2005 @07:15:14 PM EDT)
• Mayank Kabra (02.01.2005 @12:36:33 AM EDT)
• Rustem Aidagulov (02.01.2005 @01:05:30 AM EDT)
• IQSTAR (02.01.2005 @02:01:48 AM EDT)
• Narendra Agrawal (02.01.2005 @04:40:40 AM EDT)
• Dan Dima (02.01.2005 @07:14:10 AM EDT)
• Raicho Tchalkov (02.01.2005 @07:16:09 AM EDT)
• Kenneth Chan (02.01.2005 @07:16:10 AM EDT)
• Jose H. Nieto (02.01.2005 @08:41:31 AM EDT)
• Ivan Golovach (02.01.2005 @10:25:31 AM EDT)
• Joe BGI SF (02.01.2005 @01:20:54 PM EDT)
• Fendel (02.01.2005 @01:20:54 PM EDT)
• Dinesh Layek (02.01.2005 @03:50:44 AM EDT)
• Dion K Harmon (02.01.2005 @04:00:45 PM EDT)
• Leonardo Liang (02.02.2005 @02:06:22 AM EDT)
• Hao Wu (02.02.2005 @02:20:51 AM EDT)
• Dharmadeep Muppalla (02.02.2005 @03:46:09 AM EDT)
• Igor Malashkin (02.02.2005 @03:47:33 AM EDT)
• Dr. Luke Pebody (02.02.2005 @05:19:51 AM EDT)
• Guangzhong Sun (02.02.2005 @06:35:19 AM EDT)
• Vineet Dwivedi (02.02.2005 @12:43:06 PM EDT)
• Yunpeng Zhou (02.02.2005 @12:51:46 PM EDT)
• Hagen von Eitzen (02.02.2005 @03:55:31 PM EDT)
• Vineet Gupta (02.02.2005 @04:12:14 PM EDT)
• Vincent Vermaut (02.02.2005 @04:29:12 PM EDT)
• Ken Crounse (02.02.2005 @04:49:03 PM EDT)
• Ashwin Lall (02.02.2005 @05:12:59 PM EDT)
• Piros Lucian (02.02.2005 @07:29:47 PM EDT)
• Kennedy (02.02.2005 @08:00:30 PM EDT)
• Kalyana Chakravarthy (02.02.2005 @08:30:19 PM EDT)
• Bart De Vylder (02.03.2005 @07:15:19 AM EDT)
• David Friedman (02.03.2005 @11:53:43 AM EDT)
• Christian Vanhulle (02.03.2005 @12:40:35 PM EDT)
• Vinko Marinkovic (02.03.2005 @09:22:00 AM EDT)
• Sachin Shukla (02.04.2005 @03:51:45 AM EDT)
• Victor Chang (02.04.2005 @04:16:37 AM EDT)
• Omer Faruk Alis (02.04.2005 @04:47:36 AM EDT)
• Dipak K Jha (02.05.2005 @12:47:33 PM EDT)
• Bhaskara Aditya (02.04.2005 @06:41:38 AM EDT)
• Pascal Brisset (02.04.2005 @09:53:25 AM EDT)
• Tameen Khan (02.04.2005 @11:03:22 AM EDT)
• Amit Jain (02.04.2005 @02:13:00 PM EDT)
• Emma Rebato (02.04.2005 @02:22:22 PM EDT)
• Anand (02.04.2005 @03:18:57 PM EDT)
• John Hart (02.04.2005 @04:31:50 PM EDT)
• Daniel Bitin (02.04.2005 @04:43:48 PM EDT)
• George Ciobanu (02.04.2005 @06:28:35 PM EDT)
• Samir Kagadkar (02.04.2005 @06:39:36 PM EDT)
• Coserea Gheorghe (02.04.2005 @07:38:29 PM EDT)
• Alexey Vorobyov (02.04.2005 @08:33:04 PM EDT)
• Kim Se Kwon (02.04.2005 @11:31:38 PM EDT)
• Dave Dodson (02.04.2005 @11:54:30 PM EDT)
• Rubén Hernández Aza (02.05.2005 @07:30:53 AM EDT)
• Lizhao Zhang (02.05.2005 @06:19:48 PM EDT)
• Pradheep Elango (02.06.2005 @01:40:55 AM EDT)
• Oskar Sigvardsson (02.06.2005 @07:07:47 PM EDT)
• Sunil Nandihalli (02.07.2005 @12:12:20 AM EDT)
• Shubin Hu (02.07.2005 @01:17:16 AM EDT)
• John G. Fletcher (02.07.2005 @01:22:58 AM EDT)
• Arup Salui (02.07.2005 @02:08:26 AM EDT)
• John Tromp (02.07.2005 @10:02:57 AM EDT)
• Scott Hotes (02.07.2005 @03:45:58 PM EDT)
• Dave Biggar (02.08.2005 @05:09:00 AM EDT)
• Guler Amos (02.08.2005 @10:23:57 AM EDT)
• Claudio Baiocchi (02.08.2005 @12:04:38 PM EDT)
• Natalia Kolganova (02.08.2005 @02:44:00 PM EDT)
• Raghav Bhaskar (02.09.2005 @10:02:22 AM EDT)
• Caltech Math Students (02.09.2005 @04:40:56 PM EDT)
• John Ritson (02.09.2005 @10:58:28 PM EDT)
• Thiru Sampath (02.09.2005 @11:02:00 PM EDT)
• Divyesh Dixit (02.10.2005 @05:35:53 AM EDT)
• Steve Powell (02.10.2005 @08:15:45 AM EDT)
• Maxim Zabara (02.11.2005 @02:32:23 AM EDT)
• Emile Joubert (02.11.2005 @10:53:42 AM EDT)
• Vince Lynch (02.11.2005 @01:14:18 PM EDT)
• David McQuillan (02.11.2005 @05:28:42 PM EDT)
• Florbela Isabel Carrasqueiras (02.12.2005 @01:10:21 PM EDT)
• Bill Webb (02.12.2005 @02:55:24 PM EDT)
• Boris Silantiev (02.12.2005 @04:33:29 PM EDT)
• Daniel Chong Jyh Tar (02.12.2005 @11:18:18 PM EDT)
• Herbert Groscot (02.13.2005 @08:05:32 AM EDT)
• Lovro Puzar (02.13.2005 @01:15:37 PM EDT)
• Karl D'Souza (02.13.2005 @05:33:51 PM EDT)
• Franco Bassi (02.13.2005 @06:19:41 PM EDT)
• Tim Lewis (02.16.2005 @04:57:36 PM EDT)
• Nathaniel Litton (02.17.2005 @01:34:00 PM EDT)
• Aditya Utturwar (02.18.2005 @10:02:43 AM EDT)
• Narayanan Rajamanin (02.19.2005 @07:08:19 PM EDT)
• Sergey Povetkin (02.20.2005 @01:40:21 PM EDT)
• Jay Lewandowski (02.21.2005 @01:18:48 AM EDT)
• Ajay Agrawal (02.24.2005 @05:16:08 AM EDT)
• Richard A Bauer (02.24.2005 @09:28:58 PM EDT)
• John Dalbec (02.26.2005 @12:06:55 AM EDT)
• Yakov Sirotkin (02.26.2005 @03:26:05 AM EDT)
• Sridevi Parise & Naval Verma (02.28.2005 @02:28:07 AM EDT)
• Petru Budur (02.28.2005 @05:43:08 AM EDT)