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Maximum Square

Problem

What are the dimensions of the shaded square inside the 3-4-5 right angle triangle?


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Solution

Let length of the side of the square be, a.


By similar triangles, (3minusa)/a = a/(4minusa)

therefore (3minusa)(4minusa) = a2

a2 minus 7a + 12 = a2

7a = 12 implies a = 12/7

Find the position of the square that has the maximum area.
Can you generalise for an x-y-z right angle triangle?
What would be the dimensions of the square in terms of x, y, and z?

Problem ID: 152 (Feb 2004)     Difficulty: 2 Star