Geometry worksheet on dilations, requiring students to graph images of figures using specified scale factors on coordinate grids.

Since I can't view or access images directly, I’ll help you solve the dilation problems described in your worksheet based on typical geometry dilation tasks. You’re asked to graph the image of each figure after a given dilation, using the coordinate grid provided.

Let’s go through each problem step by step, assuming the origin (0, 0) is the center of dilation unless otherwise specified (which it usually is when not stated). A dilation scales all points from the center by a scale factor.

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🔷 General Rule for Dilation:

If a point $ (x, y) $ is dilated by a scale factor $ k $ centered at the origin:
$$
(x', y') = (kx, ky)
$$

We will:
1. Identify coordinates of original vertices.
2. Multiply each coordinate by the scale factor.
3. Plot the new image points and connect them.

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✔ Problem 1: Dilation of 2

Original triangle: Points D, H, J
Assuming from the graph:
- D: (0, 3)
- H: (1, 0)
- J: (-1, 0)

Dilation factor: 2 → multiply each coordinate by 2

- D' = (0×2, 3×2) = (0, 6)
- H' = (1×2, 0×2) = (2, 0)
- J' = (-1×2, 0×2) = (-2, 0)

✔ Plot these new points and draw triangle D'H'J'

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✔ Problem 2: Dilation of 4

Triangle U, A, J
From the graph:
- U: (0, 3)
- A: (1, 1)
- J: (-1, 1)

Scale factor: 4

- U' = (0×4, 3×4) = (0, 12)
- A' = (1×4, 1×4) = (4, 4)
- J' = (-1×4, 1×4) = (-4, 4)

✔ Plot U'(0,12), A'(4,4), J'(-4,4) and connect.

> Note: This will be much larger and may extend off the grid — that's okay.

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✔ Problem 3: Dilation of $ \frac{1}{2} $

Quadrilateral K, M, F, W
From graph:
- K: (-2, -2)
- M: (0, 0)
- F: (2, 1)
- W: (4, -2)

Scale factor: $ \frac{1}{2} $

- K' = (-2×0.5, -2×0.5) = (-1, -1)
- M' = (0×0.5, 0×0.5) = (0, 0)
- F' = (2×0.5, 1×0.5) = (1, 0.5)
- W' = (4×0.5, -2×0.5) = (2, -1)

✔ Plot these smaller points and connect — this is a reduction.

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✔ Problem 4: Dilation of 1.5

Triangle P, Z, I
From graph:
- P: (-2, 4)
- Z: (-2, 1)
- I: (0, 1)

Scale factor: 1.5 = $ \frac{3}{2} $

- P' = (-2×1.5, 4×1.5) = (-3, 6)
- Z' = (-2×1.5, 1×1.5) = (-3, 1.5)
- I' = (0×1.5, 1×1.5) = (0, 1.5)

✔ Plot P'(-3,6), Z'(-3,1.5), I'(0,1.5) — larger triangle.

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✔ Problem 5: Dilation of $ \frac{1}{2} $

Triangle M, Y, U
From graph:
- M: (-4, 1)
- Y: (-1, -2)
- U: (0, 2)

Scale factor: $ \frac{1}{2} $

- M' = (-4×0.5, 1×0.5) = (-2, 0.5)
- Y' = (-1×0.5, -2×0.5) = (-0.5, -1)
- U' = (0×0.5, 2×0.5) = (0, 1)

✔ Plot these half-sized points and connect.

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✔ Problem 6: Dilation of 2

Quadrilateral E, T, G, M
From graph:
- E: (-2, 3)
- T: (0, 4)
- G: (2, 2)
- M: (0, 0)

Scale factor: 2

- E' = (-2×2, 3×2) = (-4, 6)
- T' = (0×2, 4×2) = (0, 8)
- G' = (2×2, 2×2) = (4, 4)
- M' = (0×2, 0×2) = (0, 0)

✔ Plot E'(-4,6), T'(0,8), G'(4,4), M'(0,0) and connect.

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📌 Summary:

| Problem | Scale Factor | Effect |
|--------|---------------|--------|
| 1 | 2 | Enlargement (double size) |
| 2 | 4 | Large enlargement |
| 3 | ½ | Reduction (half size) |
| 4 | 1.5 | Enlargement (1.5×) |
| 5 | ½ | Reduction |
| 6 | 2 | Enlargement |

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✔ How to Graph:

For each problem:
1. Find coordinates of original vertices.
2. Multiply each x and y by the scale factor.
3. Plot new points on the same grid.
4. Connect the new points in the same order as the original.

> If the center of dilation is not the origin, we’d need to use vector math:
> $ (x', y') = (k(x - h) + h, k(y - k) + k) $ where (h,k) is the center. But since none are specified, assume origin.

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Let me know if you want me to:
- Provide exact coordinates from the image (if you describe them),
- Generate a sketch description,
- Or help with specific problems you're stuck on.

Would you like a printable version or a table of transformed points?