Question

1. which transformation of the line ( x = 3 ) results in an image that is perpendicular to the given line? a. ( r_{x - axis} ) b. ( r_{y - axis} ) c. ( r_{y = x} ) d. ( r_{x = 1} )

Explanation:

Step1: Analyze the original line

The line \(x = 3\) is a vertical line. A line perpendicular to a vertical line is a horizontal line.

Step2: Analyze each transformation

• For \(r_{x - axis}\) (reflection over the \(x\) - axis):

A reflection over the \(x\) - axis of the line \(x = 3\) is still the line \(x = 3\) (a vertical line).

• For \(r_{y - axis}\) (reflection over the \(y\) - axis):

The reflection of the line \(x = 3\) over the \(y\) - axis is the line \(x=-3\) (a vertical line).

• For \(r_{y = x}\) (reflection over the line \(y = x\)):

The reflection of a point \((x,y)\) over the line \(y = x\) is \((y,x)\). For the line \(x = 3\) (points of the form \((3,y)\)), after reflection over \(y=x\), the points become \((y,3)\), and the equation of the image is \(y = 3\) (a horizontal line).

• For \(r_{x = 1}\) (reflection over the line \(x = 1\)):

The distance between \(x = 3\) and \(x = 1\) is \(|3 - 1|=2\). The reflection of \(x = 3\) over \(x = 1\) is \(x=1-(3 - 1)=-1\) (a vertical line).

Answer:

C. \(r_{y=x}\)