Chapter 12.6, Problem 3C

To determine

To find: the quadrant the image lie after a ${90}^{\circ }$ clockwise rotation and ${90}^{\circ }$ counterclockwise rotation when figure lies in third quadrant.

Answer to Problem 3C

The image lie in fourth quadrant after ${90}^{\circ }$ clockwise rotation and lies in second quadrant after ${90}^{\circ }$ counterclockwise rotation when figure lies in third quadrant.

Explanation of Solution

Given information :

Figure lies in third quadrant.

Calculation : in ${90}^{\circ }$ clockwise rotation switch the coordinates then multiply the new y -coordinate by -1.

  $\left(x,y\right)\to \left(y,-x\right)$

Because figure lies in third quadrant so its coordinate = $\left(-x,-y\right)$

After ${90}^{\circ }$ clockwise rotation,

  $\left(-x,-y\right)\to \left(-y,x\right)$

So after ${90}^{\circ }$ clockwise rotation the image lies in fourth quadrant.

In ${90}^{\circ }$ counterclockwise rotation switch the coordinates, then multiply the new x -coordinate by -1 .

  $\left(x,y\right)\to \left(-y,x\right)$

Because figure lies in third quadrant so its coordinate = $\left(-x,-y\right)$

After ${90}^{\circ }$ counterclockwise rotation,

  $\left(-x,-y\right)\to \left(y,-x\right)$

So after ${90}^{\circ }$ counterclockwise rotation the image lies in second quadrant.

Hence,

Thus, The image lie in fourth quadrant after ${90}^{\circ }$ clockwise rotation and lies in second quadrant after ${90}^{\circ }$ counterclockwise rotation when figure lies in third quadrant.