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In this problem, we are given two shapes, A and B, on a coordinate plane. Shape A is transformed to shape B by first reflecting across the x-axis, followed by a translation.

Let's break this down step by step:

### 1. **Reflection Across the X-axis**
When a shape is reflected across the x-axis, the \( y \)-coordinates of all the points change signs, but the \( x \)-coordinates remain the same. This means that shape A will have the same \( x \)-coordinates as shape B, but the \( y \)-coordinates of A will be the negative of those of B.

- Shape A is positioned in the first quadrant, while shape B is in the third quadrant. Since reflection only changes the sign of the \( y \)-coordinates, the positions relative to the \( x \)-axis confirm this reflection.

### 2. **Translation**
After the reflection, shape A needs to be translated by a vector \( \begin{pmatrix} c \\ d \end{pmatrix} \) to move it to the position of shape B.

- To find the translation vector, we pick a corresponding point from shape A and shape B (after reflection). Let's consider the top-right vertex of shape A, which is at \( (4, 6) \). After reflecting over the x-axis, this point becomes \( (4, -6) \).

- The same vertex in shape B is located at \( (-4, -2) \).

### 3. **Find Translation Vector**
Now, we find the translation from the reflected point \( (4, -6) \) to the point \( (-4, -2) \) on shape B.

- The horizontal translation (\( c \)) is the change in the \( x \)-coordinate:  
  \( c = -4 - 4 = -8 \).

- The vertical translation (\( d \)) is the change in the \( y \)-coordinate:  
  \( d = -2 - (-6) = -2 + 6 = 4 \).

### Final Answer:
- \( c = -8 \)
- \( d = 4 \)

Would you like any further clarification or details on this solution?

Here are 5 related questions to explore:

1. How would the transformation change if the reflection was over the y-axis instead?
2. Can you find another pair of corresponding points to verify the translation vector?
3. What would happen if the reflection occurred first on the y-axis and then on the x-axis?
4. How would you describe a rotation that results in shape B from shape A?
5. What would the translation be if no reflection was applied?

**Tip:** When reflecting a shape across an axis, only the coordinates perpendicular to the axis change signs.

Shape A can be transformed to shape B by a reflection in the x-axis followed by a translation. Find the value of c and the value of d.

Solve a geometry problem involving reflections across the x-axis and translations on the coordinate plane. Find the values of c and d that transform Shape A to Shape B. This solution covers key geometry concepts suitable for students in Grades 8-10.

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