Chapter 12, Problem 20CT

To determine

To draw : A $△ABC$ with vertices $A\left(-3,4\right)$ , $B\left(-2,5\right)$ and $C\left(-1,3\right)$ . Find the coordinates of the vertices of the image after the specified transformation and draw the image.

Answer to Problem 20CT

The coordinates of the vertices of the image after specified transformation are $A^{\prime }\left(-6,8\right)$ , $B^{\prime }\left(-4,10\right)$ , $C^{\prime }\left(-2,6\right)$ .

Explanation of Solution

Given information : Coordinates of the vertices $△ABC$ are $A\left(-3,4\right)$ , $B\left(-2,5\right)$ and $C\left(-1,3\right)$ .

Type of transformation: Dilation having a scale factor of $2$ .

Graph :

The graph is of $△ABC$

Concept used: In dilation, a figure stretches or shrinks with respect to the origin.

For dilation having scale factor k: $\left(x,y\right)\to (kx,ky)$

Let, after the dilation having scale factor $2$ the new points for A, B and C will be $A^{\prime }$ , $B^{\prime }$ and $C^{\prime }$

Therefore,

  $\begin{array}{l}A\left(-3,4\right)\to \left(-3\times 2,4\times 2\right)=A^{\prime }\left(-6,8\right)\\ B\left(-2,5\right)\to \left(-2\times 2,5\times 2\right)=B^{\prime }\left(-4,10\right)\\ C\left(-1,3\right)\to \left(-1\times 2,3\times 2\right)=C^{\prime }\left(-2,6\right)\end{array}$

The coordinates of the vertices of the image after specified transformation are $A^{\prime }\left(-6,8\right)$ , $B^{\prime }\left(-4,10\right)$ , $C^{\prime }\left(-2,6\right)$ .

Interpretation : The coordinates of the vertices of the image after dilation having scale factor $2$ are $A^{\prime }\left(-6,8\right)$ , $B^{\prime }\left(-4,10\right)$ , $C^{\prime }\left(-2,6\right)$ .