Question

describe the reflection that maps \\(overline{st}\\), with vertices \\(s(-7, 1)\\) and \\(t(-7, 5)\\), onto itself. use decimals, if necessary. reflection in the line \\(y =\\)

Explanation:

Step1: Identify the type of line segment

The coordinates of \( S \) are \((-7, 1)\) and \( T \) are \((-7, 5)\). Since the \( x \)-coordinates are the same (\( x=-7 \)), the line segment \( \overline{ST} \) is a vertical line segment.

Step2: Find the midpoint of \( \overline{ST} \)

For a vertical line segment with endpoints \((x, y_1)\) and \((x, y_2)\), the midpoint \( M \) has coordinates \((x, \frac{y_1 + y_2}{2})\). Here, \( x=-7 \), \( y_1 = 1 \), and \( y_2 = 5 \). So the midpoint \( M \) is \((-7, \frac{1 + 5}{2})\).

Step3: Calculate the midpoint's \( y \)-coordinate

Calculate \( \frac{1+5}{2}=\frac{6}{2} = 3 \). The line of reflection that maps a vertical segment onto itself is the horizontal line passing through its midpoint. Since the segment is vertical (parallel to the \( y \)-axis), the line of reflection is horizontal (parallel to the \( x \)-axis) and has the equation \( y = 3 \), because reflecting over this line will map each point on the segment to another point on the segment (since it's the midline of the vertical segment).

Answer:

\( 3 \)