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Equal Areas of Pictures and Frames

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Two pictures have equal area. Their frames, which have a thickness of \(1\) inch, also have equal area.

Prove that the two frames are identical.

Note by Daniel Liu
2 years, 6 months ago

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First, cut out all the square \(1\) inch corners. Then if \(a,b\) and \(c,d\) are the dimensions of the pictures, we then see that the perimeters and areas have to be the same. The rest follows. Michael Mendrin · 2 years, 6 months ago

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@Michael Mendrin Prove that if the perimeters and areas are equal, then the rectangles must be congruent. Daniel Liu · 2 years, 6 months ago

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@Daniel Liu If a+b=c+d and ab=cd, then a-b=c-d, therefore a=c and b=d.

I'll try to think of a more clever proof, if nobody else improves on this.

Okay, maybe I'll post a graphic a bit later to illustrate this. Given 2 rectangles of equal areas. Overlap them so that they share a common corner. Then, excluding the union of the 2 rectangles, we have 2 smaller rectangles, the areas of which have to be the same. If the perimeters of both rectangles are the same, then both of the smaller rectangles have to have a side of the same length. This means that the other sides they have must also be of the same length. This means that the 2 rectangles are congruent.

Picture Frames

Picture Frames

Michael Mendrin · 2 years, 6 months ago

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@Michael Mendrin Technically \(a-b=\pm(c-d)\) which gives \(a=c,b=d\) or \(a=d,b=c\). Daniel Liu · 2 years, 6 months ago

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@Daniel Liu yeah something like that Michael Mendrin · 2 years, 6 months ago

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@Michael Mendrin What geometry diagram-making tool do you use? It looks a little tacky.

I recommend downloading Geogebra, it's free the quality is very good. Daniel Liu · 2 years, 6 months ago

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@Daniel Liu I used colored pencils and then I put the picture into my Underwood typewriter and clacked on some letters and then I scanned it and put it here. Just now downloaded and tried out Geogebra. Maybe I'll get the hang of it one day. Michael Mendrin · 2 years, 6 months ago

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