tag:blogger.com,1999:blog-8231784566931768362.post1997554412508063726..comments2018-01-13T06:51:48.298-05:00Comments on MathNotations: Inscribed Square in Right Triangle InvestigationDave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-8231784566931768362.post-66019825057801780162008-11-10T08:36:00.000-05:002008-11-10T08:36:00.000-05:00Hi Dave,Yes, My problem (1) was trying to show tha...Hi Dave,<BR/><BR/>Yes, My problem (1) was trying to show that there were an infinity of triangles of area two that inscribe the square of area 1. <BR/><BR/>I found the solution to my problem (2) somewhat surprising. According to my calculations, square 1 is larger than square 2 if h < (sqrt(2)-1)*c<BR/><BR/>TCTotally_cluelesshttps://www.blogger.com/profile/06449079338919787252noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-61824909571874659222008-11-08T07:17:00.000-05:002008-11-08T07:17:00.000-05:00Well, I see it now that I've worked through more o...Well, I see it now that I've worked through more of the details and my brain is actually functioning in the AM! Yes, there are 'many' such triangles whose area is 2. One can show that they all have a base of 2 and a height of 2. Of course the isos. rt. triangle is one special case of this, indeed a 'limiting' case of the side of the square lying on a side of the triangle, That is, we can slide the side of the square toward one vertex of the triangle until it reaches the 'corner'.<BR/><BR/>Alex and tc --<BR/>Thank you for enriching the original problem.Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-65293819781102994272008-11-07T15:25:00.000-05:002008-11-07T15:25:00.000-05:00Nice generalizations, tc...The first one is straig...Nice generalizations, tc...<BR/>The first one is straightforward assuming you want the result in terms of c and h. <BR/><BR/>If one inscribes the 1x1 square in an isosceles right triangle so that a side of the square lies on the hypotenuse, then the area of the square is 9/4 or 2.25. This is actually a nice problem for students to try, although not difficult. <BR/><BR/>On the other hand, if the 1x1 square is placed in the "corner" of the isos. right triangle so that only one vertex of the square is on the hypotenuse, the area of the triangle is 2 as previously noted. <BR/><BR/>Alex, tc--<BR/>Could you describe for us another triangle containing this square whose area is 2? <BR/>tc, is that what you were trying to do with your questions?Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-8422318391083814142008-11-06T20:38:00.000-05:002008-11-06T20:38:00.000-05:00Some further generalizations:1. Consider triangle ...Some further generalizations:<BR/><BR/>1. Consider triangle ABC with angles A and B acute. Construct a square that has its base on side AB of length c and two other vertices on sides AC and BC. Let the altitude from C to AB be of length h. Find the length of the side of the square.<BR/><BR/>2. Now, triangle ABC is isosceles, with angles A & B acute, but C is obtuse. Let the length of side AB be equal to c, and let the altitude from C to AB be of length h. You can have square 1 like in the previous problem. Alternatively, you can have square 2 within the triangle whose diagonal is the altitude from C to AB. Find the relationship between c & h so that the area of square 1 is greater than the area of square 2.<BR/><BR/>TCTotally_cluelesshttps://www.blogger.com/profile/06449079338919787252noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-7305916484122550772008-11-06T19:27:00.000-05:002008-11-06T19:27:00.000-05:00tc--I'm not sure if it's unique, but if we are loo...tc--<BR/>I'm not sure if it's unique, but if we are looking for the triangle of smallest area, I think I have a proof that this area is 2. The other issue is orientation of the square inside the triangle. I am assuming, perhaps incorrectly, that we are restricting our attention to triangles in which at least one side of the square lies on a side of the triangle. With this condition, my proof demonstrates that the minimum area is 2. Let me echo tc's comment: Nice question! It deserves to be published as a separate challenge.Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-12674539375762512142008-11-06T16:01:00.000-05:002008-11-06T16:01:00.000-05:00Hi Dave,I don't think that the isosceles right tri...Hi Dave,<BR/><BR/>I don't think that the isosceles right triangle is the unique solution for #3, though its area might be equal to the area of the smallest triangle.<BR/><BR/>Nice problem, Alex!<BR/><BR/>TCTotally_cluelesshttps://www.blogger.com/profile/06449079338919787252noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-19587552875277408512008-11-05T23:03:00.000-05:002008-11-05T23:03:00.000-05:00For #3, Alex, I'm leaning toward the isos. right t...For #3, Alex, I'm leaning toward the isos. right triangle of area 2 as the 'smallest'. I think I can prove it but I'm wondering if my conjecture is right and if you have a proof in mind before I suggest an argument for it.Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-50814779968923630232008-11-05T16:00:00.000-05:002008-11-05T16:00:00.000-05:00Nice, Alex!Your first question reminds me of a pos...Nice, Alex!<BR/>Your first question reminds me of a post I published last year. Look<BR/><A HREF="http://mathnotations.blogspot.com/2007/03/problems-3-5-thru-3-6-07-geometry-and.html" REL="nofollow">here.</A><BR/><BR/>#2 looks straightforward using bases and heights. #3 I really like. I'm assuming you mean the triangle of smallest area.Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-63150906584290734882008-11-05T15:25:00.000-05:002008-11-05T15:25:00.000-05:00Related questions? Hmm...1) Find the side length ...Related questions? Hmm...<BR/><BR/>1) Find the side length of the biggest square you can fit in an equilateral triangle<BR/><BR/>2) Prove that the biggest right-angled triangle you can fit in a square is the obvious one<BR/><BR/>3) What's the smallest triangle you can draw that contains a 1x1 square?Alexhttps://www.blogger.com/profile/12912173255354626589noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-83333869886050318812008-10-31T16:59:00.000-04:002008-10-31T16:59:00.000-04:00Ah,I think I was trying to say something like 'Wha...Ah,<BR/><BR/>I think I was trying to say something like 'What is the ratio of the sides a to b that maximizes the side of the square?'<BR/><BR/>Do we need to put some kind of a normalization condition on a and b? Not clear to me, but I don't have the time to think right now. <BR/><BR/><BR/>TCTotally_cluelesshttps://www.blogger.com/profile/06449079338919787252noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-72281782187720418602008-10-31T16:50:00.000-04:002008-10-31T16:50:00.000-04:00tc--Am I missing something here? There is only one...tc--<BR/>Am I missing something here? There is only one 'inscribed'square for each right triangle. Are you referring to finding the rectangle of maximum area for a given right triangle? OR??<BR/>DaveDave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-58526054510866554872008-10-31T16:20:00.000-04:002008-10-31T16:20:00.000-04:00Taking it one step further, when is the side of th...Taking it one step further, when is the side of the square maximized?<BR/><BR/>One way is to use the AM:HM (rather than AM:GM) inequality.<BR/><BR/>TCTotally_cluelesshttps://www.blogger.com/profile/06449079338919787252noreply@blogger.com