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# Proof that a scaling factor of a scales a polygon's area by a²? (Due in 3 hours)

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Our teacher asked us to find and prove the scaling of a polygon's area when you scale it by a factor of a. I've searched for a long while, and I've found answers saying that it's a², but there is no proof for it. All I've found so far is a mention of this saying that it's hard to prove, or visual interpretations using a rectangle/triangle, but no general n-gon. I have no clue where to start since there is not a general formula for any n-gon. Could anyone help me out?

Mar 23, 2022

#1
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Let X be a square with side length 3 and A be 4. Scaling by a, we have a 12 x 12 square now. The area of this square is 12 x 12 = 144, \(4^2\) times bigger than the original square's area with 9. Thus, scaling by a increases the area by \(a^2\). (Note: If you still don't believe me, try this out with a different shape and different values for X and A.

Mar 23, 2022
#2
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I'm not sure what scaling means, but if it means what I think it means, consider the following.

Take a square, with sides S.  The area is (S • S) = S2.

Scale it by a.  Now the sides are (S • a) and the area is (S • a) • (S • a) = S2a2.

Make it a rectangle, scale by a, and the areas is (L • a) • (W • a) = LWa2

Try a circle of radius R, scale by a, and the area is π • (Ra)2 = πR2a2

Mar 23, 2022
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Try watching this.

I ahve watch 3/4 of it and it looks really good.

Remember that If 2 figures are just different by a scaling factor, then they are similar figures.

Mar 23, 2022
#4
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Thanks.  I did watch it, and I know why you quit 3/4 of the way through, LOL.  :-)

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Guest Mar 24, 2022
edited by Guest  Mar 24, 2022
#5
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ok thanks for responding.  Did you get anything from it?

I quit 3/4 of the way through because I already know that stuff.

It was not the most exciting presentation but the content was good.

Melody  Mar 24, 2022
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< Did you get anything from it? >

Yes, I did.  It was really good.  He confirmed what I had kinda figured out – I've taken higher math than that, but my courses never addressed polygon expansion – and he presented it in an understandable way.  He repeats himself a lot, but that's better than not explaining enough.  Thanks again for the link.

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Guest Mar 25, 2022
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Thanks for the feedback.  :)

You can take this one step further.

For any three dimensional object:

If you expand the lengths by a factor of y ,  [so a side of lenth 4 is epanded to length 4y]

then the area of any surface will become y^2 times bigger

and the volume will enlarge by    y^3

This is useful for many physics applications too.

Melody  Mar 25, 2022