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:
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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.
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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 \( 9 - m \).
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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).
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Find the y-coordinate of the line of reflection: Setting \( m = \frac{9 + (-2)}{2} = \frac{7}{2} \), we can reflect the point \( (-3, 9) \).
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Compute reflection for \( X \): If \( m = y \) where the vertical distance \( 9 - \frac{7}{2} = \frac{11}{2} \), then \( X' \) has the formula \( Y' = -2 + \frac{11}{2} = \frac{7}{2} - 9 = -\frac{11}{2}. \)
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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).
1 answer