Geometry Transformation Composition Key - Free Printable

Let's solve the problem step by step.

We are given a pre-image with points:

- D(9, -3)
- E(6, -7)
- F(3, -3)
- G(5, -1)

And a composition of transformations:

$$
R_{x=2} \circ T_{180^\circ} \circ T_{\langle -10, -1 \rangle}
$$

This means we apply the transformations in order from right to left:

1. Translation: $ T_{\langle -10, -1 \rangle} $
2. Rotation: $ T_{180^\circ} $ (rotation about the origin)
3. Reflection: $ R_{x=2} $ (reflection over the vertical line $ x = 2 $)

We will apply these transformations step-by-step to each point.

---

Step 1: Apply $ T_{\langle -10, -1 \rangle} $

This translation moves every point by $ (-10, -1) $:
$$
(x, y) \to (x - 10, y - 1)
$$

Apply to each point:

- D(9, -3) → D' = (9 - 10, -3 - 1) = (-1, -4)
- E(6, -7) → E' = (6 - 10, -7 - 1) = (-4, -8)
- F(3, -3) → F' = (3 - 10, -3 - 1) = (-7, -4)
- G(5, -1) → G' = (5 - 10, -1 - 1) = (-5, -2)

So after first transformation, we have:
- D'(-1, -4)
- E'(-4, -8)
- F'(-7, -4)
- G'(-5, -2)

---

Step 2: Apply $ T_{180^\circ} $ — Rotation 180° about the origin

A 180° rotation about the origin transforms:
$$
(x, y) \to (-x, -y)
$$

Apply to each:

- D'(-1, -4) → D'' = (1, 4)
- E'(-4, -8) → E'' = (4, 8)
- F'(-7, -4) → F'' = (7, 4)
- G'(-5, -2) → G'' = (5, 2)

Now we have:
- D''(1, 4)
- E''(4, 8)
- F''(7, 4)
- G''(5, 2)

---

Step 3: Apply $ R_{x=2} $ — Reflection over the line $ x = 2 $

To reflect a point $ (x, y) $ over the vertical line $ x = 2 $, use the formula:

$$
x' = 2 \cdot 2 - x = 4 - x
$$
$ y $ stays the same.

So:
$$
(x, y) \to (4 - x, y)
$$

Apply to each:

- D''(1, 4) → D''' = (4 - 1, 4) = (3, 4)
- E''(4, 8) → E''' = (4 - 4, 8) = (0, 8)
- F''(7, 4) → F''' = (4 - 7, 4) = (-3, 4)
- G''(5, 2) → G''' = (4 - 5, 2) = (-1, 2)

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Final Answer:

Now fill in the table:

| Order of Transformations | Coordinates |
|--------------------------|-------------|
| $ T_{\langle -10, -1 \rangle} $ | |
| D' | (-1, -4) |
| E' | (-4, -8) |
| F' | (-7, -4) |
| G' | (-5, -2) |
| $ T_{180^\circ} $ | |
| D'' | (1, 4) |
| E'' | (4, 8) |
| F'' | (7, 4) |
| G'' | (5, 2) |
| $ R_{x=2} $ | |
| D''' | (3, 4) |
| E''' | (0, 8) |
| F''' | (-3, 4) |
| G''' | (-1, 2) |

---

Summary of Final Images:

- D'''(3, 4)
- E'''(0, 8)
- F'''(-3, 4)
- G'''(-1, 2)

You can now plot these on the grid if needed.

✔ Answer Complete.