The orientation of a figure is reversed after a reflection. The orientation of a figure is the same after a translation.
D
Use the side lengths of △CDE.
E
If the two figures are congruent, their sizes and shapes are the same.
A
B
CD=3units, DE=4units, EC=5units
C
See solution.
D
C'D'=3units, D'E'=4units, E'C'=5units
E
Yes, see solution.
Practice makes perfect
We know that triangle CDE has vertices at (1,4), (1,1), and (5,1). We can graph △ CDE by plotting the points and connecting them.
We want to find the lengths of the sides of △ CDE. We can see from the graph that the length of CD is 3 units and the length of DE is 4 units. We also know that angle CDE is a right angle.
Since this is a right triangle, let's apply the Pythagorean Theorem to find the side length of EC.
CD^2+DE^2=EC^2
We will substitute the values into the formula and solve for EC.
Since EC is the side length of the triangle and lengths cannot be negative, we know that the length of the side must be positive. We can write the side lengths of △ CDE as follows.
CD=3units
DE=4units
EC=5units
Let's reflect △ CDE over the y-axis. The image of the reflection will be △ C''D''E''.
The side lengths of the triangles are the same but the orientation of the triangle is reserved after the reflection. Next, we can translate △ C''D''E'' two units to the left.
Now we want to find the lengths of the sides of △ C'D'E'. Let's remember the properties of the reflection and translation.
Translation
Reflection
Lengths are the same.
Lengths are the same.
Orientations are the same.
Orientations are reserved.
We know that the side lengths of figures after a reflection and a translation do not change. We found the side lengths of the preimage △ CDE in Part B. We can use these lengths to write the lengths of the sides of △ C'D'E'.
CD=3units DE=4units EC=5units ⇒ C'D'=3units D'E'=4units E'C'=5units
We are asked to determine whether △ CDE is congruent to △ C'D'E'. We found a series of the transformations that maps △ CDE onto △ C'D'E' in Part C. We also know that the sizes and shapes of the triangles are the same, so these two triangles — △ CDE and △ C'D'E' — are congruent.
