Question

the coordinates of trapezoid abcd are a(-4,5), b(1,5), c(1,2), and d(-6,2). trapezoid abcd is the image after the composition r_{x - axis} ° r_{y = x} is performed on trapezoid abcd. state the coordinates of trapezoid abcd. the use of the set of axes below is optional.

Explanation:

Step1: Find the rule for the composition of reflections

The rule for reflection over the line \(y = x\) is \((x,y)\to(y,x)\).
The rule for reflection over the \(x -\)axis is \((x,y)\to(x, - y)\).
So, for the composition \(r_{x - axis}\circ r_{y=x}\), if we start with a point \((x,y)\), first apply \(r_{y = x}\): \((x,y)\to(y,x)\), then apply \(r_{x - axis}\): \((y,x)\to(y,-x)\)

Step2: Apply the rule to each vertex

• For vertex \(A(-4,5)\):

First, apply \(r_{y = x}\): \((-4,5)\to(5,-4)\)
Then, apply \(r_{x - axis}\): \((5,-4)\to(5,4)\)

• For vertex \(B(1,5)\):

First, apply \(r_{y = x}\): \((1,5)\to(5,1)\)
Then, apply \(r_{x - axis}\): \((5,1)\to(5,-1)\)

• For vertex \(C(1,2)\):

First, apply \(r_{y = x}\): \((1,2)\to(2,1)\)
Then, apply \(r_{x - axis}\): \((2,1)\to(2,-1)\)

• For vertex \(D(-6,2)\):

First, apply \(r_{y = x}\): \((-6,2)\to(2,-6)\)
Then, apply \(r_{x - axis}\): \((2,-6)\to(2,6)\)

Answer:

\(A''(5,4)\), \(B''(5,-1)\), \(C''(2,-1)\), \(D''(2,6)\)