McGraw Hill Glencoe Geometry, 2012
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McGraw Hill Glencoe Geometry, 2012 View details
7. Congruence Transformations
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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 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 and 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 and as well as and have the same coordinates, To find the measure of and we can find the difference of the coordinates of the endpoints of each segment.
The segments have the same measures, which means that and are congruent segments. Similarly, we can calculate the measures of and However, this time we calculate the difference between the coordinates of the endpoints of the segments.
Segments and have also the same measures and are congruent segments. Finally, to calculate the measures of and we will use the Distance Formula.
Let's substitute the coordinates of the endpoints of and into the formula.
and and

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