Question

graph each pair of the following lines and for each construct the corresponding line of reflection so that the image of one line in the pair will be the second line. for each, identify the line of reflection

a. ( y = -x ) and ( y = x )
b. ( y = 10x ) and ( y = -10x )

what are the equations of the lines of reflection?

\\( \bigcirc \\) a. ( y = 1 ) and ( x = 1 )
\\( \bigcirc \\) b. ( y = 10x ) and ( y = 10x )
\\( \bigcirc \\) c. ( x = 0 ) and ( y = 0 )
\\( \bigcirc \\) d. ( y = x ) and ( y = -x )

Explanation:

Step1: Analyze the reflection of \(y = -x\) and \(y=x\)

The line \(y = x\) and \(y=-x\) are symmetric about the \(x -\)axis (\(y = 0\)) and \(y -\)axis (\(x = 0\)). For a point \((a,b)\) on \(y=-x\) (\(b=-a\)), its reflection about the \(x -\)axis is \((a, - b)=(a,a)\) which lies on \(y = x\) and vice - versa. Similarly, for a point \((a,b)\) on \(y = 10x\) (\(b = 10a\)), its reflection about the \(y -\)axis is \((-a,b)=(-a,10(-a))\) which lies on \(y=-10x\) and vice - versa.

Step2: Check each option

• Option A: \(y = 1\) and \(x = 1\) are horizontal and vertical lines not passing through the origin. The lines \(y=-x,y = x,y = 10x,y=-10x\) all pass through the origin \((0,0)\). So, they cannot be lines of reflection.
• Option B: \(y = 10x\) and \(y = 10x\) are the same line. This does not represent lines of reflection.
• Option C: The \(x -\)axis (\(y = 0\)) is the line of reflection for \(y=-x\) and \(y = x\) (since if \((x,y)\) is on \(y=-x\) (\(y=-x\)), then \((x,-y)\) is on \(y = x\) (\(-y=x\)) and vice - versa). Also, for \(y = 10x\) and \(y=-10x\), if \((x,y)\) is on \(y = 10x\) (\(y = 10x\)), then \((-x,y)\) is on \(y=-10x\) (\(y=-10(-x)\)). The \(y -\)axis (\(x = 0\)) is the line of reflection.
• Option D: \(y=x\) and \(y=-x\) are the given lines (for part a) not the lines of reflection for both pairs of lines.

Answer:

C. \(x = 0\) and \(y = 0\)