Dilation Of Obejcts On The Coordinate Plane

📝 Practice Worksheets

Dilation: Identify Scale Factor Dilation: Scale Factor Practice Dilation: Center Not Origin Dilation: About the Origin Dilation: Fractional Scale Factor

🛠️ Related Tools

Dilation Explorer

🔑 Key Concepts

A dilation enlarges or reduces a figure while keeping the same shape.
A scale factor greater than \(1\) creates an enlargement.
A scale factor between \(0\) and \(1\) creates a reduction.
The center of dilation controls where the image is located.
Lines through corresponding points meet at the center of dilation.
Corresponding sides of the preimage and image are parallel and proportional.

✏️ Worked Examples

Card 1: What a Dilation Does

A dilation changes the size of a figure.

\(k>1\Rightarrow\text{enlargement}\)

\(0<k<1\Rightarrow\text{reduction}\)

If \(k=2\), every distance from the center of dilation doubles.

If \(k=\displaystyle \frac{1}{2}\), every distance from the center of dilation is cut in half.

Card 2: Center at the Origin

When the center of dilation is the origin, you can multiply each coordinate by the scale factor.

Example: Dilate \(A(4,4)\) by \(k=2\).

\(A(4,4)\rightarrow A'(2\cdot4,2\cdot4)\)

\(A'=(8,8)\)

The point moves twice as far from the origin, but it stays on the same ray from the origin.

Card 3: Lines of Dilation

Corresponding points lie on the same straight line from the center of dilation.

For example, \(A\) and \(A'\), \(B\) and \(B'\), and \(C\) and \(C'\) should line up with the center.

If those lines all meet at one point, that point is the center of dilation.

\( \text{center}\rightarrow A\rightarrow A' \)

This is a useful way to check whether the dilation was drawn correctly.

Card 4: Distances Are Proportional

The line from the center to a preimage point and the line from the center to its image point are proportional.

If the scale factor is \(2\), then:

\( \text{center to }A'=2(\text{center to }A) \)

If the scale factor is \(\displaystyle \frac{1}{2}\), then:

\( \text{center to }A'=\displaystyle \frac{1}{2}(\text{center to }A) \)

So lines of dilation can help show how far the image points should be from the center.

Card 5: Center Not at the Origin

In the video, figure \(WXYZ\) is dilated by \(k=0.5\) using \((6,2)\) as the center of dilation.

Because \(0.5=\displaystyle \frac{1}{2}\), the image will be smaller.

The distance from \((6,2)\) to \(W\) is \(10\) units.

\(\displaystyle \frac{1}{2}\cdot10=5\)

So \(W'\) is placed \(5\) units from the center, on the same line from the center to \(W\).

Card 6: Use Corresponding Sides

After locating one image point, corresponding sides can help build the rest of the image.

In the original figure, \(WZ=6\) units.

Since \(k=\displaystyle \frac{1}{2}\), the image side is:

\(W'Z'=\displaystyle \frac{1}{2}(6)=3\)

So draw \(W'Z'\) parallel to \(WZ\), but only \(3\) units long.

All corresponding sides stay parallel and are multiplied by the same scale factor.

🧠 Math Vocabulary

Dilation: a transformation that enlarges or reduces a figure while keeping the same shape.
Scale Factor: the number \(k\) that tells how much larger or smaller the image is compared to the preimage.
Enlargement: a dilation with a scale factor greater than \(1\).
Reduction: a dilation with a scale factor between \(0\) and \(1\).
Center of Dilation: the fixed point where lines through corresponding points meet.
Image: the new figure after the dilation.
Preimage: the original figure before the dilation.
Corresponding Points: matching points on the preimage and image, such as \(A\) and \(A'\).
Corresponding Sides: matching side lengths in the preimage and image.
Similar Figures: figures with the same shape and proportional side lengths.
Non-Rigid Transformation: a transformation that changes the size of a figure.

💡 Main Idea

A dilation creates a similar image by multiplying distances from the center of dilation by a scale factor. The image may be larger or smaller, and corresponding points should line up with the center of dilation.

📚 What You Should Already Know

Students should know how to plot points on the coordinate plane, multiply coordinates by whole numbers and fractions, identify corresponding sides, and recognize proportional relationships.

🚀 What Comes Next

Next, students can practice dilations with different centers of dilation, compare enlargements and reductions, and connect dilations to similarity, scale drawings, and proportional side lengths.

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