Chapter 12, Problem 27CR

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 27CR

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

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: ${180}^{\circ }$ rotation.

Graph :

The graph is of $△PQR$

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

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

Let, after the ${180}^{\circ }$ rotation the new points for P, Q and R will be $P^{\prime }$ , $Q^{\prime }$ and $R^{\prime }$

Therefore,

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

The coordinates of the vertices of the image after ${180}^{\circ }$ clockwise rotation are $P^{\prime }\left(-1,-2\right)$ , $Q^{\prime }\left(-2,-3\right)$ , $R^{\prime }\left(-3,-1\right)$ .

Interpretation : The coordinates of the vertices of the image after ${180}^{\circ }$ clockwise rotation are $P^{\prime }\left(-1,-2\right)$ , $Q^{\prime }\left(-2,-3\right)$ , $R^{\prime }\left(-3,-1\right)$ .