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Prove that the △XYZ and △LKJ are congruent triangles.
Reflection, see solution.
We are asked to identify the type of transformation and prove that it is a congruence 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-axis. Therefore, the diagram shows the reflection.
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.
XZ | LJ |
---|---|
X(-9,4) and Z(-1,-3) | L(9,4) and J(1,-3) |
XZ=(-1−(-9))2+(-3−4)2 | LJ=(1−9)2+(-3−4)2 |
XZ=(-1+9)2+(-3−4)2 | LJ=(-8)2+(-7)2 |
XZ=82+(-7)2 | LJ=64+49 |
XZ=64+49 | LJ=113 |
XZ=113 | LJ=10.630145… |
XZ=10.630145… | LJ≈10.6 |
XZ≈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.