Chapter 12.6, Problem 32E

i.

To determine

To find the coordinates of the vertices of $\Delta ABC\text{,}\Delta A\text{'}B\text{'}C\text{'}$ , draw the image of $\Delta A\text{'}B\text{'}C\text{'}$ and $\Delta A\text{''}B\text{''}C\text{''}$ .

Answer to Problem 32E

The coordinates of vertices of $\Delta ABC$ after ${90}^{\circ }$ clockwise rotation is $A\text{'}\left(2,6\right),B\text{'}\left(5,2\right),C\text{'}\left(1,4\right)$ .

The coordinates of vertices of $\Delta A\text{'}B\text{'}C\text{'}$ after reflection in the x-axis is $A\text{'}\left(2,-6\right),B\text{'}\left(5,-2\right),C\text{'}\left(1,-4\right)$ .

The image of $\Delta A\text{'}B\text{'}C\text{'}$ after ${90}^{\circ }$ clockwise rotation is:

  

After the reflection of $\Delta A\text{'}B\text{'}C\text{'}$ on x-axis, the image of $\Delta A\text{''}B\text{''}C\text{''}$ is:

  

Explanation of Solution

Given:

The coordinates of vertices of $\Delta ABC$ is $A\left(-6,2\right),B\left(-2,5\right),C\left(-4,1\right)$ .

Calculation:

First draw

  $\Delta ABC$ . Then to rotate

  $\Delta ABC{90}^{\circ }$ clockwise, switch the coordinates and multiply the new

  $y$ -coordinate by $-1$ .

Original	Image
$\Delta ABC$ (x,y)	$\Delta A\text{'}B\text{'}C\text{'}$ (x,y)
$A\left(-6,2\right)$	$A\text{'}\left(2,6\right)$
$B\left(-2,5\right)$	$B\text{'}\left(5,2\right)$
$C\left(-4,1\right)$	$C\text{'}\left(1,4\right)$

Therefore,

The coordinates of vertices of $\Delta ABC$ after ${90}^{\circ }$ clockwise rotation is $A\text{'}\left(2,6\right),B\text{'}\left(5,2\right),C\text{'}\left(1,4\right)$ .

The image of $\Delta A\text{'}B\text{'}C\text{'}$ after ${90}^{\circ }$ clockwise rotation is:

  

The coordinates of vertices of $\Delta A\text{'}B\text{'}C\text{'}$ is $A\text{'}\left(2,6\right),B\text{'}\left(5,2\right),C\text{'}\left(1,4\right)$ .

Then reflecting

  $\Delta A\text{'}B\text{'}C\text{'}$ on the x-axis,

First draw

  $\Delta A\text{'}B\text{'}C\text{'}$ . Then, to reflect

  $\Delta A\text{'}B\text{'}C\text{'}$ in the x-axis, multiply the y-coordinate of each vertex by $-1$ .

Original	Image
$\Delta A\text{'}B\text{'}C\text{'}$ (x,y)	$\Delta A\text{''}B\text{''}C\text{''}$ (x,y)
$A\left(2,6\right)$	$A\text{'}\left(2,-6\right)$
$B\left(5,2\right)$	$B\text{'}\left(5,-2\right)$
$C\left(1,4\right)$	$C\text{'}\left(1,-4\right)$

Therefore,

The coordinates of vertices of $\Delta A\text{'}B\text{'}C\text{'}$ after reflection in the x-axis is $A\text{'}\left(2,-6\right),B\text{'}\left(5,-2\right),C\text{'}\left(1,-4\right)$ .

After the reflection of $\Delta A\text{'}B\text{'}C\text{'}$ on x-axis, the image of $\Delta A\text{''}B\text{''}C\text{''}$ is:

  

The coordinates of vertices of $\Delta A\text{''}B\text{''}C\text{''}$ is $A\text{'}\left(2,-6\right),B\text{'}\left(5,-2\right),C\text{'}\left(1,-4\right)$ .

ii.

To determine

To answer and explain the statement.

Answer to Problem 32E

Yes.

The image remains the same, since the size, shape and angle of the image remains the same only the position of the image might change. Therefore, if the order of the transformation switch it does not change the image.

Explanation of Solution

Given:

The image of $\Delta A\text{''}B\text{''}C\text{''}$ is:

  

Yes.

The image remains the same, since the size, shape and angle of the image remains the same only the position of the image might change. Therefore, if the order of the transformation switch it does not change the image.