Scale Factors and Dilations of Polygons (B) Worksheet | 8th Grade ... - Free Printable

Let's solve this step-by-step. The task is to dilate each shape using the given scale factor and center of dilation (marked with an "X"). We'll go through each section one by one.

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🔷 Understanding Dilation with Negative Scale Factors

- A negative scale factor means the image is:
- Flipped (reflected) across the center of dilation.
- Scaled by the absolute value of the factor.
- For example:
- Scale factor = $-1$: The image is the same size but reflected over the center.
- Scale factor = $-2$: The image is twice as large and reflected.
- Scale factor = $-\frac{1}{2}$: The image is half the size and reflected.

We will use the grid to measure distances from the center of dilation (X) to each vertex, then apply the scale factor.

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## ✔ Section 1: Scale Factor = -1

Shapes: A and B

Shape A (Green Triangle)

- Center of dilation: Green X
- Scale factor = -1 → Reflection over the center

#### Step-by-step:
1. Identify the coordinates of the vertices of triangle A.
- Let’s assume grid units are 1 square.
- Suppose:
- Bottom-left: (2, 4)
- Top-left: (2, 6)
- Bottom-right: (4, 4)
2. Center of dilation (green X): approximately at (5, 6)

Now, for each point, find the vector from the center to the point, reverse it (since scale factor = -1), and plot.

But since scale factor = -1, the image will be a point reflection — each point is the same distance from the center but in the opposite direction.

So:
- For each vertex of A, draw a line from the center through the vertex, extend it the same distance beyond the center.

After doing that:
- The image of A will be a triangle symmetric to A about the green X.
- It will be the same size but flipped.

✔ Draw the reflected triangle on the other side of the green X, maintaining equal distances.

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Shape B (Yellow Triangle)

- Center of dilation: Orange X near (10, 7)
- Scale factor = -1 → Again, reflection

Steps:
1. Find coordinates of B:
- Let’s say:
- Left: (10, 8)
- Right: (14, 8)
- Top: (12, 10)
2. Center X: (10, 7)

Now, for each vertex:
- Vector from X to vertex → multiply by -1 → new position.

Example:
- Vertex (10,8): vector from (10,7) is (0,1) → ×(-1) = (0,-1) → new point: (10,6)
- Vertex (14,8): vector (4,1) → ×(-1) = (-4,-1) → new point: (6,6)
- Vertex (12,10): vector (2,3) → ×(-1) = (-2,-3) → new point: (8,4)

So, the image of B is a triangle with vertices at:
- (10,6), (6,6), (8,4)

✔ Draw this reflected triangle below and left of the orange X.

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## ✔ Section 2: Scale Factor = -2

Shapes: C and D

This means:
- Image is twice as large and reflected over the center.

Shape C (Blue Triangle)

- Center of dilation: Blue X (around (6,9))
- Scale factor = -2

#### Steps:
1. Find coordinates of C:
- Assume:
- Left: (4,9)
- Top: (6,11)
- Right: (8,9)
2. Center X: (6,9)

Vectors from center to each vertex:
- (4,9): vector = (-2,0) → ×(-2) = (4,0) → new point: (6+4, 9+0) = (10,9)
- (6,11): vector = (0,2) → ×(-2) = (0,-4) → new point: (6,5)
- (8,9): vector = (2,0) → ×(-2) = (-4,0) → new point: (2,9)

So image has vertices:
- (10,9), (6,5), (2,9)

✔ This is a larger triangle, reflected over the blue X, pointing downward.

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Shape D (Orange Parallelogram)

- Center of dilation: Red X (around (12,10))
- Scale factor = -2

Find coordinates of D:
- Assume:
- Bottom-left: (10,9)
- Bottom-right: (14,9)
- Top-left: (11,11)
- Top-right: (15,11)

Center X: (12,10)

Compute vectors:
- (10,9): vector = (-2,-1) → ×(-2) = (4,2) → new point: (12+4,10+2) = (16,12)
- (14,9): vector = (2,-1) → ×(-2) = (-4,2) → (12-4,10+2) = (8,12)
- (11,11): vector = (-1,1) → ×(-2) = (2,-2) → (12+2,10-2) = (14,8)
- (15,11): vector = (3,1) → ×(-2) = (-6,-2) → (12-6,10-2) = (6,8)

New vertices:
- (16,12), (8,12), (14,8), (6,8)

✔ Plot these points and connect to form a larger, reflected parallelogram.

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## ✔ Section 3: Scale Factor = -½

Shapes: E and F

This means:
- Half the size, and reflected over the center.

Shape E (Purple Arrowhead)

- Center of dilation: Purple X (around (5,18))
- Scale factor = -½

Assume coordinates of E:
- Points forming the arrowhead:
- Top: (7,20)
- Left: (5,18)
- Right: (9,18)

Center X: (5,18)

Vectors:
- (7,20): vector = (2,2) → ×(-½) = (-1,-1) → new point: (5-1,18-1) = (4,17)
- (5,18): vector = (0,0) → stays (5,18)
- (9,18): vector = (4,0) → ×(-½) = (-2,0) → (3,18)

So image vertices:
- (4,17), (5,18), (3,18)

✔ Smaller triangle, reflected, pointing down-left.

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Shape F (Red Arrow)

- Center of dilation: Pink X (around (15,18))
- Scale factor = -½

Shape F is more complex — a rectangle with a triangle on the right.

Let’s identify key points:

Assume:
- Rectangle: from (8,16) to (12,18) → so corners:
- (8,16), (12,16), (12,18), (8,18)
- Arrow tip: (14,17) and (14,19)? Wait, better to trace:
- From (12,18) to (14,17), (14,19), and back?

Actually, from the image:
- Arrow base: (8,16) to (12,16), up to (12,18), then to (14,17), (14,19), and back to (12,18)? Not quite.

Better:
- Main rectangle: (8,16) to (12,18)
- Arrowhead: from (12,18) to (14,17), (14,19), and back to (12,18)

So vertices:
- (8,16), (12,16), (12,18), (14,17), (14,19)

Center X: (15,18)

Now compute vectors from (15,18):

1. (8,16): vector = (-7,-2) → ×(-½) = (3.5,1) → new point: (15+3.5, 18+1) = (18.5,19)
2. (12,16): vector = (-3,-2) → ×(-½) = (1.5,1) → (16.5,19)
3. (12,18): vector = (-3,0) → ×(-½) = (1.5,0) → (16.5,18)
4. (14,17): vector = (-1,-1) → ×(-½) = (0.5,0.5) → (15.5,18.5)
5. (14,19): vector = (-1,1) → ×(-½) = (0.5,-0.5) → (15.5,17.5)

So new points:
- (18.5,19), (16.5,19), (16.5,18), (15.5,18.5), (15.5,17.5)

✔ These are half-size and reflected across the pink X.

Since the grid is in whole squares, you may need to estimate halfway points.

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## 📌 Summary of What to Do

For each shape:
1. Identify the center of dilation (X).
2. Measure vector from X to each vertex.
3. Multiply by the scale factor.
4. Plot the new point.
5. Connect the new points.

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## ✔ Final Answer: How to Complete the Worksheet

You should now:
- Reflect and scale each shape accordingly:
- Scale factor -1: Flip over center, same size.
- Scale factor -2: Flip and double size.
- Scale factor -½: Flip and halve size.

Use the grid to ensure accuracy.

> ⚠️ Since I can't draw here, your job is to:
> - Use a ruler to draw lines from the center X through each vertex.
> - Measure the distance and mark the new point at the scaled distance.
> - Connect the new points.

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## 🔍 Tips for Accuracy

- Always start from the center of dilation (X).
- Use coordinates if possible (e.g., count squares).
- Negative scale factors mean the image is on the opposite side of the center.
- The image must be similar (same shape) but possibly different size and orientation.

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✔ You've now learned how to perform dilations with negative scale factors! Practice makes perfect.

Let me know if you'd like me to generate the coordinates or sketch a description of the final images.