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UNIT
14
14.4: Dilate a Figure
14
\(\triangle ABC\) is dilated by scale factor \(k\). Its image \(\triangle A’B’C’\) is dilated by scale factor \({\frac 1k}\).

The second image is \(\triangle A''B’’C’’\). The center for both dilations is \((0,\;0)\). What can you conclude about \(\triangle ABC\) and \(\triangle A’’B’’C’’\)? Explain how you arrived at your conclusion.

Key Vocabulary

Center of dilation
Enlargement
Reduction
Scale factor
Image
Preimage
Similar figures
Corresponding sides
Ratio of corresponding sides
Multiply
Divide
  • \(\triangle ABC\) and \(\triangle A′′B′′C′′\) are the same figure.
  • Since both dilations are centered at \((0,\; 0)\), the coordinate rule \((x,\; y) \longrightarrow (kx,\; ky)\) can be applied to get the image of \(A,\; B\) and \(C\).
  • Dilation of scale factor \(k\): A\((x,\; y)\longrightarrow\) A’\((kx,\; ky)\)
  • Dilation of image A’ by scale factor 1/k: A’\((kx,\; ky)\) \(\longrightarrow\) A’’ \((\frac 1k • kx, \frac 1k • ky)\)
  • A’\((kx, ky)\) \longrightarrow A’’\(( x, y)\)
  • \(A\) and \(A’’\) have the same coordinates. Therefore \(\triangle ABC\) and \(\triangle A′′B′′C′′\) are the same figure.
  • Use the coordinate rule \((x,\; y) \longrightarrow (kx,\; ky)\) where \(k\) is the scale factor.
  • Use scale factor = k for the first dilation and scale factor = 1/k for the second dilation.