Dilation Of Triangle - Enlarging And Reducing

🔑 Key Concepts

• A dilation resizes a figure using a scale factor.
• When the center of dilation is the origin, each coordinate is multiplied by the scale factor.
• A scale factor greater than 1 enlarges the figure.
• A scale factor between 0 and 1 reduces the figure.
• A dilation usually creates a similar, non-congruent image unless the scale factor is 1.

✏️ Worked Example

Reduction: Scale Factor \(k=0.5\)

Center of dilation: \((0,0)\)
Rule: \((x,y)\rightarrow(0.5x,0.5y)\)

\(A(3,3)\rightarrow A'(1.5,1.5)\)
\(B(5,3)\rightarrow B'(2.5,1.5)\)
\(C(4,5)\rightarrow C'(2,2.5)\)

Since \(0<k<1\), the image is a reduction. The image is similar to the preimage, but not congruent.

Enlargement: Scale Factor \(k=2\)

Center of dilation: \((0,0)\)
Rule: \((x,y)\rightarrow(2x,2y)\)

\(A(2,3)\rightarrow A'(4,6)\)
\(B(3,3)\rightarrow B'(6,6)\)
\(C(2.5,4.5)\rightarrow C'(5,9)\)

Since \(k>1\), the image is an enlargement. The image is similar to the preimage, but not congruent.

🧠 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 a figure is enlarged or reduced.
• Preimage: The original figure before the dilation.
• Image: The new figure after the dilation.
• Enlarge: To make a figure larger. This happens when the scale factor is greater than 1.
• Reduce: To make a figure smaller. This happens when the scale factor is between 0 and 1.
• Non-rigid transformation: A transformation that changes the size or shape of a figure.
• Congruent figures: Figures with the same size and same shape.
• Non-congruent figures: Figures that do not have the same size and shape.
• Similar figures: Figures with the same shape but not necessarily the same size.

💡 Main Idea

A dilation changes the size of a figure by multiplying coordinates or distances by a scale factor. When the center of dilation is the origin, the rule is \((x,y)\rightarrow(kx,ky)\). A scale factor greater than 1 enlarges the figure, while a scale factor between 0 and 1 reduces it.

📚 What You Should Already Know

Students should know how to plot ordered pairs, multiply positive and negative numbers, identify the origin, and understand that similar figures have the same shape.

🚀 What Comes Next

Students can extend dilations by identifying scale factors from graphs, dilating figures when the center is not the origin, and comparing dilations with translations, reflections, and rotations.

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