Equal Areas of Pictures and Frames

Imgur Imgur

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.

#Algebra #Area #Picture #ProofProblems #Equal

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 - 6 years, 9 months ago

Prove that if the perimeters and areas are equal, then the rectangles must be congruent.

Daniel Liu - 6 years, 9 months ago

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

Michael Mendrin - 6 years, 9 months ago

@Michael Mendrin – Technically $a-b=\pm(c-d)$ which gives $a=c,b=d$ or $a=d,b=c$ .

Daniel Liu - 6 years, 9 months ago

@Daniel Liu – yeah something like that

Michael Mendrin - 6 years, 9 months ago

@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 - 6 years, 9 months ago

@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 - 6 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$