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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.