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 Packing 'em In (Posted on 2004-03-02)

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 k². Similarly, triangles BPS and CQR together have area (1-k)². Thus, the area of the bottom rectangle is 1 - k² - (1-k)² = 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)k². 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)k² = k²/2. From this, the sum of the areas of F and G is k²/2 + 2k(1-k) = 2/3 - 3/2(k-2/3)². 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.

 Subject Author Date answer K Sengupta 2007-08-29 11:55:46 Another way Axorion 2004-03-03 19:27:47 re(4): We are not square (in my original post, they are) Juggler 2004-03-03 19:20:57 re(3): We are not square (in my original post, they are) DJ 2004-03-03 13:39:12 re(2): We are not square (in my original post, they are) Juggler 2004-03-03 11:54:25 re: Layman's Maths / we are not SQUARE!!! Ady TZIDON 2004-03-03 02:30:44 Layman's Maths Juggler 2004-03-03 02:07:02 The Calculus Charlie 2004-03-02 15:36:59 A generalization Federico Kereki 2004-03-02 14:43:10 re(2): Possible solution Federico Kereki 2004-03-02 14:41:02 re(2): Possible solution DJ 2004-03-02 14:25:18 re: Possible solution SilverKnight 2004-03-02 14:13:25 Possible solution Federico Kereki 2004-03-02 14:07:23

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