Question

after a reflection of the figure, the image vertices are a(5, 1), b(3, -1), and c(7, -1). what is the line of reflection? a. y = 2 b. x = 2 c. y = -x d. y = 2x

Explanation:

Step1: Observe the y - coordinates of image vertices

The y - coordinates of \(B'(3, - 1)\) and \(C'(7,-1)\) are the same (\(y=-1\)). For a reflection, we can consider the mid - point of the pre - image and image points. Since the y - coordinates of \(B'\) and \(C'\) are constant, the line of reflection is a horizontal line. Let's consider the y - coordinate of \(A'\) which is \(y = 1\).

Step2: Analyze the pre - image and image relationship for y - values

We know that for a reflection over a horizontal line \(y = k\), if a point \((x,y)\) is reflected to \((x,y')\), then \(k=\frac{y + y'}{2}\). Since the figure is symmetric about the line of reflection, and considering the nature of reflection, we can find the line of reflection by looking at the y - value of the mid - line between the original and reflected points. The line of reflection is \(y = 2\) because the distance from \(y=-1\) to \(y = 2\) is the same as the distance from \(y = 2\) to \(y=5\) (in terms of the y - coordinate of \(A'\)).

Answer:

A. \(y = 2\)