ANSWER
[tex]\begin{gathered} P(-2,3)\to P^{\prime}(1,4) \\ K(1,9)\to K^{\prime}(4,10) \\ R(5,6) \\ Yes \end{gathered}[/tex]EXPLANATION
We want to apply the given rule to find the new points:
[tex](x,y)\to(x+3,y+1)[/tex]To do this, we have to add 3 to the x coordinate and 1 to the y coordinate of the preimage to get the image of the transformation (which is a translation).
Hence, for P(-2,3), the image is:
[tex]\begin{gathered} P(-2,3)\to P^{\prime}^{}(-2+3,3+1) \\ P(-2,3)\to P^{\prime}(1,4) \end{gathered}[/tex]For K(1,9), the image is:
[tex]\begin{gathered} K(1,9)\to K^{\prime}(1+3,9+1) \\ K(1,9)\to K^{\prime}(4,10) \end{gathered}[/tex]To find the preimage, we have to find the inverse of the transformation i.e. subtract 3 from the x coordinate and 1 from the y coordinate of the image.
Hence, the preimage of R is:
[tex]\begin{gathered} R^{\prime}(8,7)\Rightarrow R(8-3,7-1) \\ \Rightarrow R(5,6) \end{gathered}[/tex]A transformation is an isometry when the shape preimage and the image of the transformation are congruent, in other words, they have the same shape and size.
The transformation above is a translation. A translation is an isometry because it moves the given shape through a fixed length in a fixed direction.
Hence, the transformation is an isometry.
