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Pythagorean Pyramid (Posted on 2004-04-26) Difficulty: 3 of 5
The pictured tetrahedron has four identical rectangular (i.e., right-angled or pythagorean) triangles as faces, with

AB=CD=p,
AC=BD=q,
AD=BC=r,
and p+q=r.

What's its volume, as a function of p, q and r?






  Submitted by Federico Kereki    
Rating: 4.1250 (8 votes)
Solution: (Hide)
The volume of such a pyramid is zero; the four points are coplanar. To see why, let's work with coordinates. Let A be at (0,0,0), B at (p,0,0), C at (0,q,0), and D at (x,y,z).

We have AD=r, so x+y+z=r; BD=q, so (x-p)+y+z=q; and CD=p, so x+(y-q)+z=p.

Subtracting BD from AD, 2px-x=p, so x=p. Subtracting CD from AD, 2qy-q=q, so y=q. Finally, substituting these values in AD, p+q+z=r, so z=0, and D lies in the same plane as A, B and C.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some ThoughtsSlaying the Hedra (Spoiler)Dej Mar2007-12-06 11:17:48
answerK Sengupta2007-05-04 14:53:58
SolutionNo Subjectnikki2004-04-28 22:47:20
re: SolutionCharlie2004-04-27 09:18:39
Questionre: SolutionCharlie2004-04-26 20:58:03
re: SolutionCharlie2004-04-26 20:55:42
Some Thoughtsre: SolutionOskar2004-04-26 17:46:50
SolutionVictor Zapana2004-04-26 16:59:56
SolutionSolution by rotatione.g.2004-04-26 15:42:41
SolutionWorking with spherese.g.2004-04-26 12:25:38
Another ViewJer2004-04-26 12:11:38
SolutionGeometricallyOskar2004-04-26 10:27:19
SolutionTwo waysCharlie2004-04-26 09:51:06
Hints/TipsIs this true?Oskar2004-04-26 09:28:51
Some ThoughtsStrange...e.g.2004-04-26 09:25:34
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