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**Karuna****Guest**

Find the image of the triangle ABC with the vertices A(2,1),B(4,3),C(3,1) under a stretch, scale factor 2. with invariant line y=x

**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,610

hi Karuna,

Welcome to the forum.

In a stretch the movement is perpendicular to the invarient line. For a scale factor of 2, points move away to twice the distance.

Hope that helps.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**rk_nithi****Novice**- Registered: 2024-10-25
- Posts: 1

Bob wrote:

hi Karuna,

Welcome to the forum.

In a stretch the movement is perpendicular to the invarient line. For a scale factor of 2, points move away to twice the distance.

https://i.imgur.com/QqaD6h5.gif

Hope that helps.

Bob

Thanks for the feedback. Answer is correct. i would like to understand how to prove this with mathematical steps or equation? also can you tell me which software you are using to plot the graph?

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,610

hi rk_nithi

Welcome to you as a new member!

Easy bit first. This is a vector graphics program called Geometer's Sketchpad. There is also a free program called Geogebra which works similarly. Geo is more versatile but takes longer to learn. Sketchpad costs money but I got it years ago (before Geo) and it still serves me well. I like the control it gives over thickness of lines and colours.

Transformations that leave the origin invariant can be represented by a matrix transformation. So, in theory there must be one for this one but I'll have to do a bit of work to find it. I'll come back to this later.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,610

OK, got it.

Are you familiar with matrix multiplication? If not then there are two useful pages here:

https://www.mathsisfun.com/algebra/matr … lying.html

https://www.mathsisfun.com/algebra/matr … lator.html

In 2D transformation geometry 2 by 1 vectors (coordinates switched row and column) can be transformed by multiplying by a 2 by 2 matrix.

As any such matrix transforms (0,0) to (0,0) this only works when the transform leaves the origin invariant. The one for the question does as y=x goes through the origin.

I'll call the matrix

y=x is invariant so (x,x) maps onto (x,x)

This gives us two equations

a + b = 1 so b = 1-a and c + d = 1 so d = 1-c

So the matrix becomes

Now to fix which stretch by considering a single point under the transformation.

I'll start on the line at (2,2) and go one right and one down to (3,1) and again to (4,0)

That's a good point to consider as there's a zero in the calculation.

It's image is at one right and one down and the same again ie at (6,-2)

So (4,0) maps onto (6,-2)

This leads to a = 1.5 and c = -0.5

So the matrix for this transformation is

So what does this do to our points?

QED

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,868

Hi Bob;

Bob wrote:

There is also a free program called Geogebra which works similarly. Geo is more versatile but takes longer to learn. Sketchpad costs money but I got it years ago (before Geo) and it still serves me well. I like the control it gives over thickness of lines and colours.

Bob

I thought I'd try Geogebra to test the comparison with your Sketchpad, and here's what I got:

I used the Classic 6 update version which I'd never used before, and found that it took a bit of learning because of the changes from my old Classic 5. Some things just weren't intuitive enough for me now, but maybe it does more...

*EDIT: Tried it again, and it took about 10 minutes. Getting more familiar with it...*

*Last edited by phrontister (2024-10-26 11:24:31)*

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,610

hi Phro,

That's excellent. I didn't know all that was possible in geo. It seems to have got a lot better since I last used it (probably 15 years ago!)

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,868

Hi Bob;

Btw, I can get exactly the same image with my 'old' (2021) Geogebra Classic 5 (Classic 6 is 2023).

My earliest use of Geogebra was in 2012, but I don't know which version or how capable it was.

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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