Triangle ABC is shown on the coordinate plane.



Triangle ABC is dilated by a scale factor of  about the origin and then reflected over the x-axis. What is the location of A″ after the sequence of transformations?

 (8, −2) (4, 1) (4, −1) (−4, 1)

Let's analyze the sequence of transformations: a dilation followed by a reflection over the x-axis. We need to determine the final position of point \( A \) after these transformations.

### Step 1: Understand the dilation
- The triangle is **dilated by a scale factor \( k \)** about the origin.
- After dilation, any point \( (x, y) \) on the triangle becomes \( (kx, ky) \).

### Step 2: Apply the reflection over the x-axis
- Reflecting a point \( (x, y) \) over the x-axis results in \( (x, -y) \).

### Let's work through these transformations step by step.

#### Assume point A is originally located at \( (x_1, y_1) \).

#### Step 3: Apply the dilation
- After dilation by a factor \( k \), the coordinates of point \( A' \) become:
  \[
  A' = (kx_1, ky_1)
  \]

#### Step 4: Apply the reflection over the x-axis
- Reflecting \( A' = (kx_1, ky_1) \) over the x-axis gives the final coordinates \( A'' \):
  \[
  A'' = (kx_1, -ky_1)
  \]

### Checking the given options:
Since the problem doesn't explicitly provide the coordinates of point \( A \) or the scale factor, I'll infer the most reasonable scenario for typical coordinate transformations.

Let's consider each of the answer options:
- **(8, −2)**: Could be a possible reflection, depending on \( k \).
- **(4, 1)**: Reflecting over the x-axis would result in a positive y-value, which doesn't match the expected reflection output.
- **(4, −1)**: Matches a typical scenario of dilation and reflection, where the x-value is positive and the y-value is negative.
- **(−4, 1)**: Reflecting over the x-axis would not result in a positive y-value for a reflected point starting in the first or third quadrant.

### Most Likely Answer:
Given typical geometric reasoning:
- The most probable location of \( A'' \), assuming a reasonable scale factor, is **\( (4, -1) \)**.

This matches the typical transformations observed for common values of scale factors and coordinate reflections.

Would you like more details or have further questions? Here are some related questions:

1. How would the answer change if the triangle was reflected over the y-axis instead?
2. What would the final position be if the dilation was about a different point, not the origin?
3. How does the scale factor affect the distances between points after dilation?
4. What are the general rules for reflecting a point over an arbitrary line, not just the x-axis?
5. Can you describe a scenario where reflecting before dilating would give a different final point?

**Tip:** Always apply transformations in the order given, as changing the order can lead to different results!

Explore how Triangle ABC is dilated by a scale factor and then reflected over the x-axis. Understand the transformations involved and determine the final position of point A′′. Suitable for high school students studying coordinate geometry and geometric transformations.

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