Chapter 11.3, Problem 16PPS

a.

To determine

To write: The co − ordinates of the vertex after following rotation-

  ${90}^{\circ }$ Clockwise about the origin

  ${180}^{\circ }$ About the origin

  ${270}^{\circ }$ Clockwise about the origin

Answer to Problem 16PPS

Rotation of ${90}^{\circ }$ clockwise about the origin gives $\left(x,y\right)\Rightarrow \left(y,-x\right)$

Rotation of ${180}^{\circ }$ about the origin gives $\left(x,y\right)\Rightarrow \left(-x,-y\right)$

Rotation of ${270}^{\circ }$ clockwise about the origin $\left(x,y\right)\Rightarrow \left(-y,x\right)$

Explanation of Solution

Given information:

Given co-ordinate of the vertices is $\left(x,y\right)$

Formula used:

Co-ordinate of the vertices is $\left(x,y\right)$, then

  ${180}^{\circ }$ rotation about the origin gives $\left(x,y\right)\Rightarrow \left(-x,-y\right)$

  ${90}^{\circ }$ clockwise about the origin gives $\left(x,y\right)\Rightarrow \left(y,-x\right)$

Co-ordinate of the vertices is $\left(x,y\right)$

After rotation of ${90}^{\circ }$ clockwise about the origin gives $\left(x,y\right)\Rightarrow \left(y,-x\right)$

After rotation of ${180}^{\circ }$ about the origin gives $\left(x,y\right)\Rightarrow \left(-x,-y\right)$

For rotation of ${270}^{\circ }$ clockwise about the origin, first rotate ${180}^{\circ }$ about the origin and then rotate ${90}^{\circ }$ clockwise about the origin.

Rotation of ${180}^{\circ }$ about the origin gives $\left(x,y\right)\Rightarrow \left(-x,-y\right)$

Then rotation of ${90}^{\circ }$ clockwise about the origin $\left(-x,-y\right)\Rightarrow \left(-y,x\right)$

Hence, after rotation of ${270}^{\circ }$ clockwise about the origin $\left(x,y\right)\Rightarrow \left(-y,x\right)$

b.

To determine

To find: The quadrant in which the vertex of the image located after rotation of ${180}^{\circ }$ about the origin.

Answer to Problem 16PPS

The vertex of the image located in fourth quadrant after rotation of ${180}^{\circ }$ about the origin.

Explanation of Solution

Given information:

The vertex of the figure is located in second quadrant.

Given that the vertex of the figure is located in second quadrant.

Let the co − ordinate of the vertex is $\left(-x,y\right)$

After rotation of ${180}^{\circ }$ about the origin gives $\left(-x,y\right)\Rightarrow \left(x,-y\right)$

This implies that after rotation of ${180}^{\circ }$ about the origin the vertex will be in fourth quadrant.

Therefore, the vertex of the image located in fourth quadrant after rotation of ${180}^{\circ }$ about the origin.