Exercise 6 Page 299

We need to identify the type of transformation and prove that it is a congruence transformation. Let's do one thing at a time.

Identify the Type of Transformation

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

We can see that each vertex and its image are in the same position, just $4$ units down and $4$ units right. Therefore, the diagram shows a translation.

Is It a Congruence Transformation?

In order to verify that this is a congruence transformation, we need to prove that $△MPS$ and $△HJK$ are congruent triangles. Let's begin with calculating the lengths of the triangle's sides. To do this, we should find the coordinates of the vertices of the triangles.

Now we can calculate the lengths of the sides of the triangles. We will substitute the coordinates of the segments' endpoints into the Distance Formula. Let's start with $\overline{MP}.$ Similarly, we can calculate the measures of $\overline{MS}$ and $\overline{PS}.$

$\overline{MS}$	$\overline{PS}$
$M(\text{-}4,7)$ and $S(\text{-}1,1)$	$P(\text{-}9,2)$ and $S(\text{-}1,1)$
$MS=\sqrt{{\left(\text{-}1-(\text{-}4)\right)}^2+{\left(1-7\right)}^2}$	$PS=\sqrt{{\left(\text{-}1-(\text{-}9)\right)}^2+{\left(1-2\right)}^2}$
$MS=\sqrt{{\left(\text{-}1+4\right)}^2+{\left(1-7\right)}^2}$	$PS=\sqrt{{\left(\text{-}1+9\right)}^2+{\left(1-2\right)}^2}$
$MS=\sqrt{9+36}$	$PS=\sqrt{64+1}$
$MS=\sqrt{45}$	$PS=\sqrt{65}$
$MS=3\sqrt{5}$	$PS\approx 8.1$

Now, using the same formula, we need to calculate the measures of the sides of $△JHK.$

$\overline{JH}$	$\overline{HK}$	$\overline{JK}$
$J(4,0)$ and $H(\text{-}1,\text{-}5)$	$H(\text{-}1,\text{-}5)$ and $K(7,\text{-}6)$	$J(4,0)$ and $K(7,\text{-}6)$
$JH=\sqrt{{\left(\text{-}1-4\right)}^2+{\left(\text{-}5-0\right)}^2}$	$HK=\sqrt{{\left(7-(\text{-}1)\right)}^2+{\left(\text{-}6-(\text{-}5)\right)}^2}$	$JK=\sqrt{{\left(7-4\right)}^2+{\left(\text{-}6-0\right)}^2}$
$JH=\sqrt{{\left(\text{-}1-4\right)}^2+{\left(\text{-}5-0\right)}^2}$	$HK=\sqrt{{\left(7+1\right)}^2+{\left(\text{-}6+5\right)}^2}$	$JK=\sqrt{(7-4{)}^2+{\left(\text{-}6-0\right)}^2}$
$JH=\sqrt{25+25}$	$HK=\sqrt{64+1}$	$JK=\sqrt{9+36}$
$JH=\sqrt{50}$	$HK=\sqrt{65}$	$JK=\sqrt{45}$
$JH=5\sqrt{2}$	$HK\approx 8.1$	$JK=3\sqrt{5}$

Let's now gather the information we have found and compare the sides of $△MPS$ and $△JHK.$ As we can see, there are three pairs of congruent segments. This allows us to conclude by the Side-Side-Side Theorem that the triangles are congruent. Therefore, the transformation is indeed a congruence transformation.