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What is the area that is the result from applying the transformation:

\begin{pmatrix}1&-\sqrt{3}\\ \sqrt{3}&1\end{pmatrix}

to a pentagon with points (-1,0), (0,1), (1,1), (0,-1), (1,0).

Jun 8, 2021

#1
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The new area is 25/2.

Jun 8, 2021
#2
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I have a passing interest in this question too.

What does it even mean?

If I was to totally guess the answer I'd say the area is 3 times that of the original.  2.5*3 = 7.5

But like I said. I have no idea what I am talking about.

Jun 13, 2021
#3
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The following should help:

Jun 13, 2021
#4
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Thanks Alan,

Did you work out the answer for this?

My logic, or lack thereof, is that the sides would increase in length by sqrt3

So the are would increase by 3

the original area was 2.5.

That is where I got 7.5 from.  Does that sound right to you?

Jun 13, 2021
#5
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Hi Melody, I make the transformed area 10.

The scaling on length is $$\sqrt{\sqrt3^2+1^2}=2$$  so the scaling on area is 22 = 4.

(Note: there's no guarantee that I know what I'm talking about either!)

Alan  Jun 13, 2021
edited by Alan  Jun 13, 2021
#6
+115901
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Thanks very much Alan.

I am sure you do know what you are talking about.

We know the original area is 2.5

We know  (-1,0) transforms to (-1,sqrt3) and   (0,1) transforms to (sqrt3,1)

We know that the distance between the original 2 points is  sqrt2 units

We know that the distance between the corresponding transformed two points is sqrt{8}

the ratio of new to old side lengths is  sqrt8/sqrt2 = 2

square it and we get the ratio of the areas is 4

The original area is 2.5 so the new area must be 10

NOW we agree :))

Melody  Jun 14, 2021
edited by Melody  Jun 14, 2021