perplexus.info :: Geometry : Rectangle-Diagonal

Rectangle-Diagonal (Posted on 2016-11-02)

Let E be the foot of the perpendicular from A to diagonal
BD in rectangle ABCD. Let F and G be points on sides BC
and CD respectively such that EFCG is a rectangle.

If |BD| = d, |EG| = g, and |EF| = 1, find d in terms of g.

See The Solution Submitted by Bractals

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Solution

Comment 1 of 1

I don't have time for a more complete solution, but this should be enough to follow:

DG=x

DC=x+1

DE=sqrt(x^2+g^2)

There are many similar triangles in the figure. Use many of them to establish:

BF=g/x

BC=g(1+x)/x

AD=(x^2+g^2)/g

Since AD=BC

x=g^(2/3)

A final proportion gives

d=BD=(g^(2/3)+1)^(3/2)

Posted by Jer on 2016-11-02 14:26:28

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