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1. Let C = (1,4), P = (5,2), and P′ = (13,−2). There is a dilation that leaves C where it is and transforms P into P′. The point C is called the dilation center. Explain why the magnitude of this dilation is 3. Calculate Q′, given that Q = (3,5). Calculate R, given that R′ = (−6, 7).

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Answer:

The dilatation is 3 because the separation between the dilatation center C and P grows by that factor of 3.

Explanation:

  1. Between C and P, the separation is |5-1| = 4 in x-axis and |2-4| = 2 in y-axis
  2. Between C and P', the separation is |13-1| = 12 in x-axis and |-2-4| = 6 in y-axis.

Comparing the separation on each direction, 12/4 = 3 and 6/2 = 3. Therefore, that's the dilatation value.

Alternatively, by computing the distances between the points

[tex]\frac{\sqrt{\left(13-1\right)^2+\left(-2-1\right)^2}}{\sqrt{\left(5-1\right)^2+\left(-2-1\right)^2}}=\frac{6\sqrt{5}}{2\sqrt{5}}=3[/tex]

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