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Packing 'em In (Posted on 2004-03-02) Difficulty: 3 of 5

ABC is an acute-angled triangle with area 1. A rectangle F(PQRS) has its vertices on the sides of the triangle, with P and Q on BC, R on AC, and S on AB. Another rectangle, G(WXYZ), has its vertices on the sides of triangle ASR, with W and X on RS, Y on AS, and Z on AR.

What is the maximum total area of F and G?

  Submitted by DJ    
Rating: 4.2000 (10 votes)
Solution: (Hide)
2/3

Let k = AR/AC. Triangle ASR is similar to ABC, with all dimensions smaller by a factor of k. Thus, its area is k.

Similarly, triangles BPS and CQR together have area (1-k). Thus, the area of the bottom rectangle is 1 - k - (1-k) = 2k(1-k). Defining h = AZ/AS in a similar way, the area of the top rectangle is 2h(1-h) times the area of the triangle ASR, or 2h(1-h)k.

Notice, now, that the area of F depends only on k, while the area of G depends on both k and h. So, the area of the top rectangle is maximized by taking h=1/2, giving G an area of 2(1/2)(1-1/2)k = k/2.

From this, the sum of the areas of F and G is k/2 + 2k(1-k) = 2/3 - 3/2(k-2/3). It is simple enough to see that this is maximized when k = 2/3, leaving the joint area of F and G to be 2/3.

It is interesting to note that this value (2/3) is the same regardless of the specific triangle.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
answerK Sengupta2007-08-29 11:55:46
Another wayAxorion2004-03-03 19:27:47
re(4): We are not square (in my original post, they are)Juggler2004-03-03 19:20:57
Some Thoughtsre(3): We are not square (in my original post, they are)DJ2004-03-03 13:39:12
Some Thoughtsre(2): We are not square (in my original post, they are)Juggler2004-03-03 11:54:25
Hints/Tipsre: Layman's Maths / we are not SQUARE!!!Ady TZIDON2004-03-03 02:30:44
Some ThoughtsLayman's MathsJuggler2004-03-03 02:07:02
SolutionThe CalculusCharlie2004-03-02 15:36:59
SolutionA generalizationFederico Kereki2004-03-02 14:43:10
Some Thoughtsre(2): Possible solutionFederico Kereki2004-03-02 14:41:02
re(2): Possible solutionDJ2004-03-02 14:25:18
Questionre: Possible solutionSilverKnight2004-03-02 14:13:25
SolutionPossible solutionFederico Kereki2004-03-02 14:07:23
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