To determine the length of the line that is reflected across the x-axis, we first need to look at the original line defined by the points \((-2, 3)\) and \((2, 3)\).
- Find the length of the line segment: The distance between the two points can be calculated using the distance formula:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For the points \((-2, 3)\) and \((2, 3)\):
- \(x_1 = -2\), \(y_1 = 3\)
- \(x_2 = 2\), \(y_2 = 3\)
Substituting the values into the formula:
\[ \text{Distance} = \sqrt{(2 - (-2))^2 + (3 - 3)^2} = \sqrt{(2 + 2)^2 + (0)^2} = \sqrt{4^2} = \sqrt{16} = 4 \]
So, the length of the original line segment is 4 units.
- Reflect the line across the x-axis: When the line is reflected across the x-axis, the y-coordinates of all points on the line will change signs, while the x-coordinates will stay the same. Therefore, the reflected points will be:
- The reflection of \((-2, 3)\) is \((-2, -3)\).
- The reflection of \((2, 3)\) is \((2, -3)\).
The reflected line segment now goes from \((-2, -3)\) to \((2, -3)\).
- Find the length of the reflected line segment: Using the same distance formula as before for the points \((-2, -3)\) and \((2, -3)\):
\[ \text{Distance} = \sqrt{(2 - (-2))^2 + (-3 - (-3))^2} = \sqrt{(2 + 2)^2 + (0)^2} = \sqrt{4^2} = \sqrt{16} = 4 \]
Thus, the length of the reflected line is also 4 units.
The answer to the question is:
4 units.