Chapter 14, Problem 8CR

(a)

To determine

To draw: The figure $\Delta ABC$ and its images under ${\Re }_{O,90}$ .

Explanation of Solution

Given information: The points are $A\left(3,2\right)$ , $B\left(-1,1\right)$ , and $C\left(1,-3\right)$ .

Graph:

Plot the given points $A\left(3,2\right)$ , $B\left(-1,1\right)$ , and $C\left(1,-3\right)$ on the coordinate plane and join the points to draw $\Delta ABC$ as shown below.

  

Figure (1)

The images of points $A$ , $B$ , and $C$ under the rotation ${\Re }_{O,90}$ are:

  $\begin{array}{c}{\Re }_{O,90}:A\left(3,2\right)\to \left(2,-3\right)\\ {\Re }_{O,90}:B\left(-1,1\right)\to \left(1,1\right)\\ {\Re }_{O,90}:C\left(1,-3\right)\to \left(-3,-1\right)\end{array}$

The image of $\Delta ABC$ under the rotation ${\Re }_{O,90}$ is shown below:

  

Figure (2)

Interpretation: From the above figures it can be observed that the figure (2) shows the rotation of all the points of $\Delta ABC$ about the origin in counterclockwise direction by ${\Re }_{O,90}$ .

(b)

To determine

To draw: The figure $\Delta ABC$ and its images under $H_O$ .

Explanation of Solution

Given information: The points are $A\left(3,2\right)$ , $B\left(-1,1\right)$ , and $C\left(1,-3\right)$ .

Graph:

Plot the given points $A\left(3,2\right)$ , $B\left(-1,1\right)$ , and $C\left(1,-3\right)$ on the coordinate plane and join the points to draw $\Delta ABC$ as shown below.

  

Figure (1)

The images of $A$ , $B$ , and $C$ under the half-turn $H_O$ is as follows.

  $\begin{array}{c}{H}_O:A\left(3,2\right)\to \left(-3,-2\right)\\ H_O:B\left(-1,1\right)\to \left(1,-1\right)\\ H_O:C\left(1,-3\right)\to \left(-1,3\right)\end{array}$

The image of $\Delta ABC$ under the half-turn $H_O$ is shown below.

  

Figure (2)

Interpretation: From the above figures it can be observed that the figure (2) shows the half-turn of $\Delta ABC$ about the origin in counterclockwise direction.