Question

A) Apply geometric motions to reflect figures. Brinda reflects point over a line of reflection,
resulting in point . What must be true? (1 point)
T (2, 3)
T (2, − 5)
′
The line of reflection is y = −1.
The line of reflection is x = −1.
The line of reflection is the x-axis.
The line of reflection is the y-axis

Answer 1

To determine what must be true about the reflection of point T(2, 3) over a line of reflection resulting in point T' (2, -5), we can analyze the coordinates given.

1. **Original Point**: T(2, 3)
2. **Reflected Point**: T'(2, -5)

To find the line of reflection, we can look at the midpoint of the segment between the original point T and the reflected point T'. The midpoint M can be calculated using:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Substituting the coordinates of points T and T':

\[ M = \left( \frac{2 + 2}{2}, \frac{3 + (-5)}{2} \right) \]
\[ M = \left( 2, \frac{3 - 5}{2} \right) \]
\[ M = \left( 2, \frac{-2}{2} \right) \]
\[ M = (2, -1) \]

Now we know the midpoint M(2, -1) lies on the line of reflection. Since T reflected over the line gives T', and the y-coordinates are differing by 8 units (3 to -5), the line of reflection must be horizontal. The y-coordinate of the mid-point, -1, indicates that the line of reflection is indeed:

\[ y = -1 \]

Therefore, the line of reflection must be \( y = -1 \).

So, the correct statement is:

**The line of reflection is \( y = -1 \).**