To find the lines of reflection for the points K and L, we can utilize the midpoint formula and the properties of reflections in geometry.
1. Line of Reflection for Point K to Point K'
Original point K: (5, 2)
Reflected point K': (-5, 2)
To find the line of reflection, we first find the midpoint (M) of segment KK': \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{5 + (-5)}{2}, \frac{2 + 2}{2} \right) = (0, 2) \]
Since the points K and K' have the same y-coordinate, the line of reflection will be a vertical line through the midpoint's x-coordinate.
Thus, the line of reflection that maps K to K' is: \[ x = 0 \]
2. Line of Reflection for Point L to Point L'
Original point L: (-3, 2)
Reflected point L': (-2, 3)
Again, we find the midpoint (M) of segment LL': \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-3 + (-2)}{2}, \frac{2 + 3}{2} \right) = \left( \frac{-5}{2}, \frac{5}{2} \right) \]
Now we look at the slope between points L and L':
- The slope of line segment LL' can be calculated as: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 2}{-2 - (-3)} = \frac{1}{1} = 1 \]
Finding the Perpendicular Line
The line of reflection will have a slope that is the negative reciprocal of the slope of LL'. Since the slope of LL' is 1, the slope of the line of reflection will be -1.
Using the midpoint (M) located at \((-2.5, 2.5)\) and the slope -1, we can use the point-slope form of a line: \[ y - y_1 = m(x - x_1) \] Substituting the values: \[ y - 2.5 = -1\left(x + 2.5\right) \]
This simplifies to: \[ y - 2.5 = -x - 2.5 \] \[ y = -x + 0 \]
Thus, the equation can be written as: \[ y = -x \]
The line of reflection that maps L to L' is: \[ y = -x \]
Summary
- The line of reflection that maps point K to K' is \(x = 0\).
- The line of reflection that maps point L to L' is \(y = -x\).
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