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The Center of Dilation

11 Jan

CCSS-M 8.G.A.4

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

1 01-10-2015 Image003

Are the figures similar?

Is one triangle a dilation of the other?

If so, where is the center of dilation?

I use the Class Capture feature of TI-Nspire Navigator to watch my students work.

2 Screen Shot 2014-11-17 at 9.30.50 AM

Who has enough information to show whether one triangle is a dilation of another?

3 Screen Shot 2014-11-17 at 9.27.06 AM4 Screen Shot 2014-11-17 at 9.27.18 AM 5 Screen Shot 2014-11-17 at 9.27.29 AM 6 Screen Shot 2014-11-17 at 9.27.43 AM 7 Screen Shot 2014-11-17 at 9.31.49 AM 8 Screen Shot 2014-11-17 at 9.32.15 AM

Whose work would you select for a whole class discussion?

9 01-10-2015 Image004

Are the triangles similar?

How do you know?

(Of course we didn’t get to this one during class … but wouldn’t you always rather have too much to do rather than too little to do?)

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5 Comments

Posted by on January 11, 2015 in Dilations, Geometry

 

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5 responses to “The Center of Dilation

  1. howardat58

    January 11, 2015 at 10:04 pm

    Taylor’s has a chance , but the one decimal place calculations of the lengths of the sloping lines is hiding the “facts”. Also, the measurements should all have been taken from the supposed centre.
    It would have been better if they had had a look at the possibilities of matching sides being parallel. Another way of looking at it is that in a dilation the line segments are dilated by the same factor. is this the case?

     
  2. howardat58

    January 11, 2015 at 10:10 pm

    Second thoughts: Take one of the joined up ones first and let them figure out that the point C’ could have been somewhere else on the middle line (might need some prodding). Then they might see that some measurement is needed.

     
    • jwilson828

      January 12, 2015 at 8:22 pm

      This is what I’m thinking about, too. How would it look for “center”, A, A be collinear’; “center”, B, B’ be collinear; and “center”, C, C’ be collinear but the figures not similar? Thanks!

       
      • howardat58

        January 20, 2015 at 1:38 pm

        The three “collinear’s” are necessary for dilation, but not sufficient (ie, not enough). Must have matching sides parallel as well. Or using O for the centre, OA’/OA = OB’/OB = OC’/OC does the job.
        If you think about dilation as expansion equally in all directions, like heating up a sheet of metal, you can see ( takes a little time !) that every point can be considered as the centre. (not of course when the dilated image is specified.

         
  3. howardat58

    January 12, 2015 at 8:58 am

    Third thoughts: Could you tie this in with scale models, and maps ?

     

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