a Note that △ XYZ is upside down compared to △ JKL.
B
b Does the composition of transformations change the shape or size of △ JKL?
A
aComposition of transformations:
Reflection: in the x-axis. Translation: (a,b) → (a+4,b)
B
b Yes, it is a congruence transformation.
Corresponding sides
Corresponding angles
XZ ≅ JL
∠ X ≅ ∠ J
XY ≅ JK
∠ Y ≅ ∠ K
YZ ≅ KL
∠ Z ≅ ∠ L
Practice makes perfect
a Examining the diagram, we notice that the orientation of △ XYZ is upside down when compared to △ JKL. Therefore, to give the triangles the same orientation, we should reflect △ JKL in the x-axis which turns it upside down. When the triangles have the same orientation, we can perform a translation to map the triangles to each other.
Reflection in the x-axis
If (a,b) is reflected in the x-axis, then its image is the point (a,- b).
preimage (a,b) → image (a,- b)
Using this rule, we can determine the coordinates of our image.
Point
(a,b)
(a,- b)
J
(- 3,2)
(- 3,- 2)
K
(- 2,4)
(- 2,- 4)
L
(0,2)
(0,- 2)
Now we can graph the image of △ JKL after reflecting it in the x-axis.
Translation
The coordinates of △ J'K'L' has the same vertical position as △ XYZ. Therefore, we only have to translate △ J'K'L' horizontally by 4 units to the right:
preimage (a,b) → image (a+4,b)
By performing this translation, we can map the triangles to each other.
The composition of transformations that maps △ JKL to △ XYZ is:
Reflection:& in the $x-$axis.
Translation:& (a,b) → (a+4,b)
b Another name for a rigid motion or a combination of rigid motions is a congruence transformation. Since both reflections and translations preserves angle and side measures, these transformations are in fact rigid motions. Therefore, we know that the composition is a congruence transformation. Let's identify the corresponding parts
&Corresponding sides &&Corresponding angles
&XZ ≅ JL && ∠ X ≅ ∠ J
&XY ≅ JK && ∠ Y ≅ ∠ K
&YZ ≅ KL && ∠ Z ≅ ∠ L
