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 reflection, we first need to identify the horizontal line of reflection.

The coordinates of point \( X \) are given as \( (-3, 9) \). The line of reflection will be located at some y-value that is midway between \( Y = 9 \) and the y-coordinate of the reflected point \( Y' \).

Given that triangle \( X'Y'Z' \) has points \( X', Y', Z' \) with \( Z' \) at \( (3, -2) \), we need to find the y-coordinate of the line of reflection. The line of reflection can be calculated as follows:

1. **Find the midpoint between \( Y \) and \( Y' \)**:
Since \( Y' \) is unknown, we can consider that the reflection will yield a coordinate of \( 9 \) above the line of reflection.
The y-coordinate of \( X' \) will be the same distance below the line as \( X \) is above it.

2. **Finding the distance from \( X \) to the line**:
Let's denote the y-coordinate of the line of reflection as \( m \). The distance from \( X \) to this line is \( 9 - m \).

3. **Setting up the equation for reflection**:
The point \( X' \) will have a y-coordinate as:
\[
Y' = m - (9 - m) = 2m - 9
\]
Since we know that the y-coordinate of \( Z' = (3,-2) \) is below the line of reflection, let's assume that the midpoint \( m \) is equal to \( -2 \) (assuming symmetry of reflection).

4. **Find the y-coordinate of the line of reflection**:
Setting \( m = \frac{9 + (-2)}{2} = \frac{7}{2} \), we can reflect the point \( (-3, 9) \).

5. **Compute reflection for \( X \)**:
If \( m = y \) where the vertical distance \( 9 - \frac{7}{2} = \frac{11}{2} \),
then \( X' \) has the formula \( Y' = -2 + \frac{11}{2} = \frac{7}{2} - 9 = -\frac{11}{2}. \)

6. Thus, the coordinates of vertex \( X' \) are:
\( (-3, -9) \) after reflection which matches with one of the options given.

Based on the steps above, the correct answer for the coordinates of vertex \( X' \) after reflection is:

**(−3,−9)**.