Question

describing types of dilations
there are distinct differences between an enlargement and a reduction with dilations. describe the key facts about both types of dilations.

Explanation:

• Enlargement (Dilation with Scale Factor > 1): When the scale factor \( k \) of a dilation is greater than 1 (\( k>1 \)), the image of the figure is larger than the original figure. For example, if a triangle has a side length of 2 units and undergoes a dilation with \( k = 3 \), the corresponding side of the image triangle will be \( 2\times3=6 \) units. The center of dilation is a fixed point, and all points of the figure are moved away from this center (or towards it but in a way that stretches the figure) such that the distance from each point to the center is multiplied by \( k \). The shape of the figure remains the same (it is similar to the original), only the size increases.
• Reduction (Dilation with Scale Factor between 0 and 1): When the scale factor \( k \) is between 0 and 1 (\( 0 < k < 1 \)), the image of the figure is smaller than the original figure. For instance, if a square has a side length of 8 units and is dilated with \( k=\frac{1}{2} \), the side length of the image square will be \( 8\times\frac{1}{2} = 4 \) units. Similar to enlargement, the center of dilation is fixed, and all points of the figure are moved closer to the center (or away from it but in a way that shrinks the figure) with the distance from each point to the center multiplied by \( k \). The figure remains similar to the original, just with a reduced size.
• Common Properties: In both cases, dilation is a similarity transformation, meaning the corresponding angles of the original and dilated figures are equal, and the ratios of corresponding sides are equal (equal to the scale factor \( k \)). The center of dilation is a crucial element as it determines the location from which the figure is scaled. Also, the orientation of the figure (e.g., the direction of vertices in a polygon) remains the same in both enlargement and reduction dilations.

Answer:

• Enlargement (Scale Factor \( k > 1 \)): The figure’s size increases (e.g., side lengths, radii) while shape (similarity) and orientation remain. Distances from the center of dilation to each point are multiplied by \( k \), stretching the figure away from (or towards, with expansion) the center.
• Reduction (Scale Factor \( 0 < k < 1 \)): The figure’s size decreases, but shape (similarity) and orientation stay. Distances from the center to each point are multiplied by \( k \), shrinking the figure toward (or away from, with contraction) the center.
• Shared Properties: Both are similarity transformations (angles equal, sides proportional), use a fixed center of dilation, and preserve shape/orientation (only size changes).