Chapter 12, Problem 19CP

a.

To determine

To rotate: Rectangle $ABCD$

Answer to Problem 19CP

The coordinates of the vertices of the image after ${180}^{\circ }$ rotation are $A^{\prime }\left(1,-2\right)$ , $B^{\prime }\left(-3,-2\right)$ , $C^{\prime }\left(-3,-4\right)$ , $D^{\prime }\left(1,-4\right)$ .

Explanation of Solution

Given information: Coordinates of the vertices of rectangle $ABCD$ are $A\left(-1,2\right)$ , $B\left(3,2\right)$ and $C\left(3,4\right)$ and $D\left(-1,4\right)$ Rectangle $ABCD$ to be rotated ${180}^{\circ }$ .

Concept used : In rotation, a figure is turned about the origin through a given angle and direction. Rectangle $ABCD$ to be rotated ${180}^{\circ }$ .

${180}^{\circ }$ rotation: $\left(x,y\right)\to \left(-x,-y\right)$

Let, after the ${180}^{\circ }$ rotation the new points for A, B, C and D will be $A^{\prime }$ , $B^{\prime }$ , $C^{\prime }$ and $D^{\prime }$ Therefore,

  $\begin{array}{l}A\left(-1,2\right)\to \left(-\left(-1\right),-\left(2\right)\right)=A^{\prime }\left(1,-2\right)\\ B\left(3,2\right)\to \left(-\left(3\right),-\left(2\right)\right)=B^{\prime }\left(-3,-2\right)\\ C\left(3,4\right)\to \left(-\left(3\right),-\left(4\right)\right)=C^{\prime }\left(-3,-4\right)\\ D\left(-1,4\right)\to \left(-\left(-1\right),-\left(4\right)\right)=D^{\prime }\left(1,-4\right)\end{array}$

The coordinates of the vertices of the image after ${180}^{\circ }$ rotation are $A^{\prime }\left(1,-2\right)$ , $B^{\prime }\left(-3,-2\right)$ , $C^{\prime }\left(-3,-4\right)$ , $D^{\prime }\left(1,-4\right)$ .

b.

To determine

To reflect: rectangle $ABCD$ in the X-axis.

Answer to Problem 19CP

Yes, $\angle 1$ and $\angle 2$ have the same measure………………..

Explanation of Solution

Given information: Coordinates of the vertices of rectangle $ABCD$ are $A\left(-1,2\right)$ , $B\left(3,2\right)$ and $C\left(3,4\right)$ and $D\left(-1,4\right)$ Rectangle $ABCD$ to be reflected in the x-axis.

Concept Used: In reflection, a figure is flipped over a line.

Reflection in X-axis: $\left(x,y\right)\to \left(x,-y\right)$

Let, after the reflection in the x-axis the new points for A, B, C and D will be ${A^{\prime }}^{\prime }$ , ${B^{\prime }}^{\prime }$ , ${C^{\prime }}^{\prime }$ and ${D^{\prime }}^{\prime }$ .

  $\begin{array}{l}A\left(-1,2\right)\to \left(-1,-\left(2\right)\right)={A^{\prime }}^{\prime }\left(-1,-2\right)\\ B\left(3,2\right)\to \left(3,-\left(2\right)\right)={B^{\prime }}^{\prime }\left(3,-2\right)\\ C\left(3,4\right)\to \left(3,-\left(4\right)\right)={C^{\prime }}^{\prime }\left(3,-4\right)\\ D\left(-1,4\right)\to \left(-1,-\left(4\right)\right)={D^{\prime }}^{\prime }\left(-1,-4\right)\end{array}$

Therefore, the coordinates of the vertices of the image after reflection in the x-axis are ${A^{\prime }}^{\prime }\left(-1,-2\right)$ , ${B^{\prime }}^{\prime }\left(3,-2\right)$ , ${C^{\prime }}^{\prime }\left(3,-4\right)$ , ${D^{\prime }}^{\prime }\left(-1,-4\right)$ .

c.

To determine

To compare: Two images of rectangle $ABCD$ .

Answer to Problem 19CP

Two images are of same size and shape but have different position.

Explanation of Solution

Given information: The images of rectangle $ABCD$ in part (a) and (b)

Rectangle $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ is the image obtained after rotating the rectangle ${180}^{\circ }$ and rectangle ${A^{\prime }}^{\prime }{B^{\prime }}^{\prime }{C}^{\prime }{D}^{\prime }$ is the image obtained by reflecting the rectangle in x-axis.

Concept used : Comparing two images just seeing the graph.

Interpretation:

Two images are of same size and shape but have different position.