Question

Use the image to answer the question.

A horizontal line separates 2 triangles. The triangle above the line is made up of points upper X (-3,9), upper Y (-2,4), and upper Z(3,4), while the triangle below the line is made up of points upper X prime, upper Y prime, and upper Z prime (3,-2)

Triangle XYZ is reflected over a horizontal line of reflection to create triangle X′Y′Z′. What are the coordinates of vertex X′?

(1 point)
Responses

(3,−9)
left parenthesis 3 comma negative 9 right parenthesis

(−3,−9)
left parenthesis negative 3 comma negative 9 right parenthesis

(−3,−7)
left parenthesis negative 3 comma negative 7 right parenthesis

(−2,−2)
left parenthesis negative 2 comma negative 2 right parenthesis
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Answer 1

To find the coordinates of vertex \( X' \) after the reflection over a horizontal line, we need to determine the horizontal distance from the point \( X(-3, 9) \) to the line of reflection, then apply the same distance below the line to find \( X' \).

Since the triangle below the line has one vertex given as \( (3, -2) \), we know the line of reflection lies at \( y = 4 \) (the average y-coordinates of points from both triangles).

1. **Find the y-coordinate of \( X' \)**: The line of reflection is \( y = 4 \), and \( X \) is located at \( y = 9 \). The distance from \( X \) to the line is:
\[
9 - 4 = 5
\]
Therefore, to find \( X' \), we subtract this distance from the line of reflection:
\[
4 - 5 = -1
\]

2. **Find the x-coordinate of \( X' \)**: The x-coordinate remains the same since this is a reflection over a horizontal line. Thus, the x-coordinate of \( X' \) is:
\[
-3
\]

Putting it all together, the coordinates of vertex \( X' \) after the reflection are:
\[
(-3, -1)
\]

However, it seems I made a mistake in considering the distance reflected; \( 9 \) is the original position beyond the line \( 4 \), so we need to reflect it correctly considering the point \( (3, -2) \) suggests we have misidentified where the line of reflection occurs.

1. Hypercorrecting as:
- Distance from \( y = 9 \) down to the line found below must yield \( y = -1 \).
- Reset the count \( 4 \) as the line split; \( 4 + x = y \) for finality.

Therefore, the corrected response for \( X' \) should be:
- Correct choices retain its form.

Adjust \( X' = (-3, -9) \), so **we select**:
**Option**: \( (-3, -9) \) as the valid vertex posture.