Exercise 5 Page 299

We are asked to identify the type of transformation and prove that it is a congruence transformation.

Identify the Type of Transformation

We are given the following diagram. Let's analyze the position of the vertices and their images to determine the type of transformation the diagram illustrates.

As we can see, each vertex and its image are the same distance from the $y\text{-}$axis. Therefore, the diagram shows the reflection.

Is It a Congruence Transformation?

To verify that this is a congruence transformation, we need to prove that $△XYZ$ and $△LKJ$ are congruent triangles. First, we need to calculate the lengths of the triangle's sides. Let's find the coordinates of the vertices of the triangles using the coordinate plane.

We can see that $X$ and $Y,$ as well as $L$ and $K,$ have the same $x\text{-}$coordinates, To find the measure of $\overline{XY}$ and $\overline{LK}$ we can find the difference of the $y\text{-}$coordinates of the endpoints of each segment. The segments have the same measures, which means that $\overline{XY}$ and $\overline{LK}$ are congruent segments. Similarly, we can calculate the measures of $\overline{YZ}$ and $\overline{JK}.$ However, this time we calculate the difference between the $x\text{-}$coordinates of the endpoints of the segments. Segments $ZY$ and $KJ$ have also the same measures and are congruent segments. Finally, to calculate the measures of $\overline{XZ}$ and $\overline{LJ},$ we will use the Distance Formula. Let's substitute the coordinates of the endpoints of $\overline{XZ}$ and $\overline{LJ}$ into the formula.

$\overline{XZ}$	$\overline{LJ}$
$X(\text{-}9,4)$ and $Z(\text{-}1,\text{-}3)$	$L(9,4)$ and $J(1,\text{-}3)$
$XZ=\sqrt{{\left(\text{-}1-(\text{-}9)\right)}^2+{\left(\text{-}3-4\right)}^2}$	$LJ=\sqrt{{\left(1-9\right)}^2+{\left(\text{-}3-4\right)}^2}$
$XZ=\sqrt{{\left(\text{-}1+9\right)}^2+{\left(\text{-}3-4\right)}^2}$	$LJ=\sqrt{{\left(\text{-}8\right)}^2+{\left(\text{-}7\right)}^2}$
$XZ=\sqrt{8^2+{\left(\text{-}7\right)}^2}$	$LJ=\sqrt{64+49}$
$XZ=\sqrt{64+49}$	$LJ=\sqrt{113}$
$XZ=\sqrt{113}$	$LJ=10.630145\dots$
$XZ=10.630145\dots$	$LJ\approx 10.6$
$XZ\approx 10.6$

As we can see, segments $XZ$ and $LJ$ are also congruent. All sides of $△XYZ$ are congruent to the corresponding sides of $△LKJ.$ Therefore, by the Side-Side-Side Theorem the triangles are congruent and hence the transformation is a congruence transformation.