To find the line of reflection between a point and its image, we can use the midpoint formula. The midpoint of the line segment connecting the original point \( Y(-3, 2) \) and its image \( Y'(2, -3) \) will lie on the line of reflection.
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Find the Midpoint: The midpoint \( M \) between the coordinates \( (-3, 2) \) and \( (2, -3) \) can be calculated using the formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Where \( (x_1, y_1) = (-3, 2) \) and \( (x_2, y_2) = (2, -3) \).
Let's calculate this: \[ M_x = \frac{-3 + 2}{2} = \frac{-1}{2} \] \[ M_y = \frac{2 + (-3)}{2} = \frac{-1}{2} \] So, the midpoint \( M \) is \( \left(-\frac{1}{2}, -\frac{1}{2}\right) \).
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Find the Slope of the Line Between \( Y \) and \( Y' \): Next, we find the slope of the line segment connecting \( Y \) and \( Y' \): \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 2}{2 - (-3)} = \frac{-5}{5} = -1 \]
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Find the Slope of the Line of Reflection: The line of reflection is perpendicular to the line connecting the original and reflected points, so its slope will be the negative reciprocal of -1, which is 1.
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Equation of the Line of Reflection: Using the slope and the midpoint, we can now use the point-slope form of the equation of a line: \[ y - y_1 = m (x - x_1) \] Here \( m = 1 \) and \( (x_1, y_1) = \left(-\frac{1}{2}, -\frac{1}{2}\right) \).
Plugging in the values: \[ y - \left(-\frac{1}{2}\right) = 1 \left(x - \left(-\frac{1}{2}\right)\right) \] Simplifying this: \[ y + \frac{1}{2} = x + \frac{1}{2} \] \[ y = x \]
Thus, the line of reflection is the line \( y = x \).