Question

apply geometric motions to reflect figures. brinda reflects point t(2, 3) over a line of reflection, resulting in point t(2, −5). what must be true? (1 point) the line of reflection is the y-axis. the line of reflection is y = −1. the line of reflection is x = −1. the line of reflection is the x-axis.

Explanation:

Step1: Recall reflection over horizontal line

For a reflection over a horizontal line \( y = k \), the \( x \)-coordinate of the point remains the same, and the \( y \)-coordinate satisfies \( k=\frac{y + y'}{2} \), where \( (x,y) \) is the original point and \( (x,y') \) is the reflected point.

Given \( T(2,3) \) and \( T'(2, - 5) \), the \( x \)-coordinate is the same (\( x = 2 \)), so it's a reflection over a horizontal line (since vertical line reflection would change \( x \)-coordinate, \( x \)-axis reflection would have \( y'=-y \), but here \( y'=-5
eq - 3\)).

Step2: Calculate the line of reflection

Using the formula for reflection over \( y = k \): \( k=\frac{y + y'}{2}=\frac{3+( - 5)}{2}=\frac{-2}{2}=-1 \). So the line of reflection is \( y=-1 \).

Let's check other options:

• Reflection over \( y \)-axis: changes \( x \)-coordinate sign, but \( x \) is 2 for both, so wrong.
• Reflection over \( x=-1 \): changes \( x \)-coordinate, but \( x \) is 2 for both, so wrong.
• Reflection over \( x \)-axis: \( y'=-y=-3

eq - 5 \), so wrong.

Answer:

The line of reflection is \( y = - 1 \). (Corresponding to the option "The line of reflection is \( y=-1 \)")