Question

On a coordinate plane, a triangle has points J (negative 3, 9), L (negative 3, 2), and K (5, 2).
Consider reflections of ΔJKL.

What line of reflection maps point K to point K' at (–5, 2)?





What line of reflection maps point L to point L' at (–2, 3)?

Answer 1

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 **\(y = -x\)**.