Chapter 12.5, Problem 20E

a.

To determine

To draw a triangle and reflect it in y-axis and reflect the image in x-axis.

Explanation of Solution

Given information:

Consider the coordinates of triangle ABC, $\text{A}\left(-5,4\right)$ , $\text{B}\left(-1,3\right)$ and $\text{C}\left(-2,1\right)$ .

Graph:

Interpretation:

The coordinate of the vertices of the final image is $\text{A''}\left(5,-4\right)$ , $\text{B''}\left(1,-3\right)$ and $\text{C''}\left(2,-1\right)$ .

b.

To determine

To draw a triangle and reflect it in x-axis and reflect the image in y-axis.

Explanation of Solution

Given information:

Consider the coordinates of triangle DEF, $\text{D}\left(0,-4\right)$ , $\text{E}\left(3,-1\right)$ and $\text{F}\left(6,-6\right)$ .

Graph:

Interpretation:

The coordinate of the vertices of the final image is $\text{D''}\left(0,4\right)$ , $\text{E''}\left(-3,1\right)$ and $\text{F''}\left(-6,6\right)$ .

c.

To determine

To perform the reflection of a figure in one axis followed by a reflection in other axis in one step.

Explanation of Solution

Given information:

Consider the coordinates of triangle ABC, $\text{A}\left(-5,4\right)$ , $\text{B}\left(-1,3\right)$ and $\text{C}\left(-2,1\right)$ .

Consider the coordinates of triangle DEF, $\text{D}\left(0,-4\right)$ , $\text{E}\left(3,-1\right)$ and $\text{F}\left(6,-6\right)$ .

Proof:

The coordinate of the vertices after reflection in y-axis followed by reflection in x-axis is $\text{A''}\left(5,-4\right)$ , $\text{B''}\left(1,-3\right)$ and $\text{C''}\left(2,-1\right)$ .

The coordinate of the vertices after reflection in x-axis followed by reflection in y-axis is $\text{D''}\left(0,4\right)$ , $\text{E''}\left(-3,1\right)$ and $\text{F''}\left(-6,6\right)$ .

Hence, To perform the reflection of a figure in one axis followed by a reflection in other axis in one step, $\left(\text{x,y}\right)\to \left(-\text{x},-\text{y}\right)$ can be used.