Moving a Triangles Center of Dilation

Shuckles

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Today my teacher threw us a challenging curve ball during geometry. The type of problem he gave us revolved around the idea that the center of dilation was the origin, but the we move it. For example one of the problems is:

Suppose we have a △ABC with coordinates as follows: A( 3,5 ); B( 1,2 ); C( 6,1 ).

Now, suppose that
△ABC undergoes a dilation of ½ about the point ( 10,10 ). How would you go about finding the coordinates for △A'B'C' with that type of scale factor? (Hint: How far would △ABC move if it were about the origin under a dilation of ½ about the origin? How far would it move towards the point ( 10,10 )?) How about a dilation with a scale factor of 3 about the point ( 3,4 )?

Basically, what would the coordinates of the 2 dilations be in the format of A( , ); B( , ); C( , )?
 
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How far is each original point from (10,10)?

Can you please be more specific? Are you talking Pythagorean Theorem? I was wondering if in the first example I add 10 onto both sides. Is that what you are getting at?
 
As a challenge my teacher … gave us …

△ABC with coordinates as follows: A( 3,5 ); B( 1,2 ); C( 6,1 ).


Now, suppose that △ABC undergoes a dilation of ½ about the point ( 10,10 ). How would you go about finding the coordinates for △A'B'C' with that type of scale factor?

(Hint: How far would △ABC move if it were … under a dilation of ½ about the origin? How far would it move towards the point (10,10 )?)


How about a dilation with a scale factor of 3 about the point ( 3,4 )?
Can you do dilations about the origin? Did you learn a formula for that, or a method graphing lines, or something else?

Did you consider the hint? That is, were you able to complete a dilation of 1/2 about the origin, followed by finding the distance? Maybe the hint suggests that a dilation about a point away from the origin might be a dilation shifted (translated) from somewhere else.
 
Can you do dilations about the origin? Did you learn a formula for that, or a method graphing lines, or something else?

Did you consider the hint? That is, were you able to complete a dilation of 1/2 about the origin, followed by finding the distance? Maybe the hint suggests that a dilation about a point away from the origin might be a dilation shifted (translated) from somewhere else.


That's what I was thinking, but the graph he gave us to graph that was 10x10, so that would go off of the paper.
 

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Can you do dilations about the origin? Did you learn a formula for that, or a method graphing lines, or something else?

Did you consider the hint? That is, were you able to complete a dilation of 1/2 about the origin, followed by finding the distance? Maybe the hint suggests that a dilation about a point away from the origin might be a dilation shifted (translated) from somewhere else.


He gave us no formula, and what you are thinking about is what I thought except the graph ends at 10,10 making it the corner. He said however, we had to graph them.
 
Can you please be more specific? Are you talking Pythagorean Theorem? I was wondering if in the first example I add 10 onto both sides. Is that what you are getting at?
You should have the Distance Formula in your pocket.

I am getting at you answering the helpful hint/question question. If you answer it, we can multiple all distances by 1/2 and get a dilation of 1/2 around (10,10) without much more trouble.
 
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