Chapter 12, Problem 26CR

To determine

To draw : A $△PQR$ with vertices $P\left(1,2\right)$ , $Q\left(2,3\right)$ and $R\left(3,1\right)$ . Find the coordinates of the vertices of the image after the specified transformation and draw the image.

Answer to Problem 26CR

The coordinates of the vertices of the image after ${90}^{\circ }$ clockwise rotation are $P^{\prime }\left(2,-1\right)$ , $Q^{\prime }\left(3,-2\right)$ , $R^{\prime }\left(1,-3\right)$ .

Explanation of Solution

Given information : Coordinates of the vertices $△PQR$ are $P\left(1,2\right)$ , $Q\left(2,3\right)$ and $R\left(3,1\right)$ .

Type of transformation: ${90}^{\circ }$ clockwise rotation.

Graph :

The graph is of $△PQR$

Concept used: In rotation, a figure is turned about the origin through a given angle and direction.

${90}^{\circ }$ Clockwise rotation: $\left(x,y\right)\to \left(y,-x\right)$

Let, after the ${90}^{\circ }$ Counterclockwise rotation the new points for P, Q and R will be $P^{\prime }$ , $Q^{\prime }$ and $R^{\prime }$

Therefore,

  $\begin{array}{l}P\left(1,2\right)\to \left(2,-1\right)=P^{\prime }\left(2,-1\right)\\ Q\left(2,3\right)\to \left(3,-2\right)=Q^{\prime }\left(3,-2\right)\\ R\left(3,1\right)\to \left(1,-3\right)=R^{\prime }\left(1,-3\right)\end{array}$

The coordinates of the vertices of the image after ${90}^{\circ }$ clockwise rotation are $P^{\prime }\left(2,-1\right)$ , $Q^{\prime }\left(3,-2\right)$ , $R^{\prime }\left(1,-3\right)$ .

Interpretation : The coordinates of the vertices of the image after ${90}^{\circ }$ clockwise rotation are $P^{\prime }\left(2,-1\right)$ , $Q^{\prime }\left(3,-2\right)$ , $R^{\prime }\left(1,-3\right)$ .