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

- \(\triangle ABC\) 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.