Infinite Algebra 2 - Dilations - Worksheets Library - Worksheets ... - Free Printable

It looks like you've uploaded a worksheet titled "Types of Dilations" that includes problems related to dilation transformations in geometry. Since I can't view images directly, I’ll guide you through how to solve typical dilation problems based on the structure and content described in your image.

Let’s go step-by-step through what this worksheet likely contains and how to approach it.

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🔍 Understanding Dilations

A dilation is a transformation that changes the size of a figure (enlarges or reduces it) while keeping its shape the same. It involves:

- A center of dilation (often the origin unless specified otherwise)
- A scale factor ($k$):
- If $|k| > 1$: enlargement
- If $0 < |k| < 1$: reduction
- If $k < 0$: the image is reflected across the center

The rule for dilation centered at the origin:
$$
(x, y) \rightarrow (kx, ky)
$$

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✔ Step-by-Step Problem Solving Guide

Let's assume the worksheet has these types of problems:

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#### Part 1: Multiple Choice Questions (Fill in the blanks)

> Example:
> 1. The scale factor of an enlargement is _________.
> 2. If the scale factor is $0.5$, the image is __________.

Answers & Explanations:

1. Greater than 1
→ Because enlargement means the image is larger than the original.

2. Reduced (or smaller)
→ Scale factor between 0 and 1 means reduction.

3. Negative
→ A negative scale factor results in a reflection across the center of dilation.

4. Enlargement
→ If $k = 2$, then each coordinate is doubled → bigger image.

5. Reduction
→ If $k = 0.75$, the image is 75% of the original size.

6. Congruent
→ Only if $k = 1$, but since it's not mentioned, maybe they're asking about similarity? Wait — actually, dilations preserve shape, so the figures are similar, not necessarily congruent.

But note: Only when $k = 1$ are they congruent.

So likely:
- Similar (not congruent) unless $k=1$

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#### Part 2: Graphing Dilation Problems

You’re given a pre-image (original shape) on a coordinate grid and asked to graph the dilated image using a scale factor and center of dilation.

Let’s suppose one problem says:

> "Dilate triangle ABC with vertices $A(2, 2)$, $B(4, 2)$, $C(3, 5)$ by a scale factor of $k = 2$ centered at the origin."

Steps:

1. Multiply each coordinate by $k = 2$:
- $A'(2×2, 2×2) = (4, 4)$
- $B'(4×2, 2×2) = (8, 4)$
- $C'(3×2, 5×2) = (6, 10)$

2. Plot the new points and connect them.

3. Label as enlargement because $k > 1$.

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Another example:

> "Dilate rectangle with vertices $(−2, −2), (2, −2), (2, 2), (−2, 2)$ by $k = 0.5$"

Then:
- New coordinates:
- $(-1, -1), (1, -1), (1, 1), (-1, 1)$
- This is a reduction (half the size)

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Now, suppose a question asks:

> "Given a triangle with vertices at $(−3, 0), (0, 3), (3, 0)$, apply a dilation with $k = −1$ centered at the origin."

Then:
- $(-3, 0) \rightarrow (3, 0)$
- $(0, 3) \rightarrow (0, −3)$
- $(3, 0) \rightarrow (−3, 0)$

This is a reflection across the origin (180° rotation), and the image is congruent to the original.

Label: Reflection or Congruent (since $|k| = 1$)

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🧩 Common Types of Dilations Based on Scale Factor

| Scale Factor ($k$) | Type | Description |
|---------------------|--------------|-----------|
| $k > 1$ | Enlargement | Image is larger |
| $0 < k < 1$ | Reduction | Image is smaller |
| $k = 1$ | Same size | No change |
| $k < 0$ | Reflection + size change | Flipped over center |

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✔ Final Tips for Answering This Worksheet

1. Identify the scale factor and center of dilation.
2. Apply the dilation formula: $(x, y) \rightarrow (kx, ky)$ if center is origin.
3. Plot the new points accurately.
4. Label the image as:
- Enlargement
- Reduction
- Congruent (only if $|k| = 1$)
- Reflected (if $k < 0$)
5. Use the terms from the key: enlargement, reduction, congruent, reflected

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💡 Example Answers (Based on Likely Questions)

Assuming the worksheet has four graphs:

1. Graph 1: Triangle scaled by $k = 2$ → enlargement
2. Graph 2: Square scaled by $k = 0.5$ → reduction
3. Graph 3: Pentagon scaled by $k = −1$ → reflected (also congruent)
4. Graph 4: Triangle scaled by $k = 3$ → enlargement

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If you can provide the specific questions or describe the figures and scale factors, I can give exact answers to your worksheet.

But based on standard worksheets like this, here’s a summary of what to write:

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✔ Summary of Answers (Likely Expected)

| Question | Answer |
|--------|--------|
| What type of dilation is $k = 2$? | Enlargement |
| What type of dilation is $k = 0.5$? | Reduction |
| What happens if $k = −1$? | Reflected (and congruent) |
| Are dilated figures always congruent? | No, only if $|k| = 1$ |
| What is the rule for dilation at origin? | $(x, y) \rightarrow (kx, ky)$ |

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Let me know if you'd like help with specific problems or want me to simulate the answers for the exact figures on your worksheet!