Finding The Scale Factor Of Dilated Shapes

🔑 Key Concepts

• A dilation uses a scale factor to enlarge or reduce a figure.
• To find the scale factor, compare a length or coordinate from the image to the matching length or coordinate from the preimage.
• If the scale factor is greater than 1, the dilation is an enlargement.
• If the scale factor is between 0 and 1, the dilation is a reduction.
• When the origin is the center of dilation, corresponding coordinates are multiplied by the same scale factor.

✏️ Worked Examples

Example 1: Larger Square to Smaller Square

The larger square is dilated to form the smaller square using the origin as the center of dilation.

Compare corresponding coordinates:
Large square point: \((6,3)\)
Smaller square point: \((2,1)\)

Use matching coordinates:
\[ \frac{2}{6}=\frac{1}{3} \] or \[ \frac{1}{3}=\frac{1}{3} \]
Scale factor: \(\displaystyle \frac{1}{3}\)

Since \(\displaystyle \frac{1}{3}<1\), this dilation is a reduction.

Example 2: Smaller Triangle to Larger Triangle

The smaller triangle is dilated to form the larger triangle using the origin as the center of dilation.

Compare corresponding coordinates:
Smaller triangle point: \((0,2)\)
Larger triangle point: \((0,6)\)

Use matching coordinates:
\[ \frac{6}{2}=3 \] Another matching pair gives:
\[ \frac{3}{1}=3 \]
Scale factor: \(3\)

Since \(3>1\), this dilation is an enlargement.

🧠 Math Vocabulary

• Dilation: A transformation that resizes a figure using a scale factor.
• Scale factor: The multiplier used to enlarge or reduce a figure.
• Center of dilation: The fixed point from which the dilation is measured.
• Preimage: The original figure before the dilation.
• Image: The new figure after the dilation.
• Enlargement: A dilation with a scale factor greater than 1.
• Reduction: A dilation with a scale factor between 0 and 1.
• Corresponding points: Points on two figures that match after a transformation.
• Similar figures: Figures with the same shape but not necessarily the same size.

💡 Main Idea

To identify the scale factor of a dilation, compare corresponding side lengths or matching coordinates. If the image is larger, the scale factor is greater than 1. If the image is smaller, the scale factor is between 0 and 1.

📚 What You Should Already Know

Students should know how to read coordinates, identify corresponding points, compare lengths, simplify fractions, and understand that dilations create similar figures.

🚀 What Comes Next

Students can use scale factors to graph dilated figures, solve dilation problems when the center is not the origin, and compare dilations with other transformations.

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