We are given the coordinates of the vertices of two triangles. Let's plot them on a coordinate plane and draw the triangles.
Now we need to analyze the position of the vertices and their images.
As we can see, each vertex and its image are the same distance from the y-axis. Therefore, by definition, the diagram illustrates a reflection.
Is it a Congruence Transformation?
In order to verify that this is a congruence transformation, we need to prove that â–ł ABC and â–ł DFG are congruent triangles.
Side Lengths of â–ł ABC
Let's begin by calculating the lengths of the sides of the triangles. We can do this by substituting the coordinates of the endpoints into the Distance Formula. Let's start with AB.
Similarly, we can calculate the measures of BC and CA.
Side
Points
Distance Formula
Side Length
AB
A( 2, 2) and B( 4, 7)
sqrt(( 4- 2)^2+( 7- 2)^2)
sqrt(29)
BC
B( 4, 7) and C( 6, 2)
sqrt(( 6- 4)^2+( 2- 7)^2)
sqrt(29)
CA
C( 6, 2) and A( 2, 2)
sqrt(( 2- 6)^2+( 2- 2)^2)
4
Side Lengths of â–ł DFG
Now, using the same formula, we can calculate the measures of the sides of â–ł DFG.
Side
Points
Distance Formula
Side Length
DF
D( 2, - 2) and F( 4, - 7)
sqrt(( 4- 2)^2+( - 7-( - 2))^2)
sqrt(29)
FG
F( 4, - 7) and G( 6, - 2)
sqrt(( 6- 4)^2+( - 2-( - 7))^2)
sqrt(29)
GD
G( 6, - 2) and D( 2, - 2)
sqrt(( 2- 6)^2+( - 2-( - 2))^2)
4
Comparing the Side Lengths
Let's now gather the information we have found and compare the side lengths of â–ł ABC and â–ł DFG.
ccc
â–ł ABC & â–ł DFG
AB=sqrt(29) & DF=sqrt(29)
BC=sqrt(29) & FG=sqrt(29)
CA=4 & GD=4
We can see that there are three pairs of congruent segments. By the Side-Side-Side Theorem, we can conclude that the triangles are congruent. Therefore, the transformation is indeed a congruence transformation.
