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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
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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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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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 invarient 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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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 invarient. The one for the question does as y=x goes through the origin.
I'll call the matrix
Now to fix which stretch by considering a singlempoint under the trasnformation.
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.
lunchtime back soon to complete.
Bob
y=x is invarient 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
lunchtime back soon to complete.
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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