a Two of the corresponding sides are horizontal or vertical. What rotations would give these sides the same orientation?
B
b What types of motions are the kind of transformations you performed to map the quadrilaterals onto each other?
A
aExample Solution:
Rotation: 90^(∘) about the origin.
Reflection:in the x-axis.
B
b Yes
Explanation: See solution.
Practice makes perfect
a From the graph, it looks like the figures have the same size so we do not have to perform any dilations to map the quadrilateral onto each other. However, they do have different positions and orientations.
Orientation
If we are mapping WXYZ to ABCD, then WZ and AD are corresponding sides. Since WZ is vertical and AD is horizontal, we can align the orientations of these sides by rotating WXYZ by 90^(∘) counterclockwise about the origin. When we rotate a figure 90^(∘) counterclockwise about the origin, the coordinates of the figures vertices change in the following way:
preimage (a,b)→ image (- b,a)
Using this rule on the vertices of WXYZ, we can determine the vertices of W'X'Y'Z'.
Point
(a,b)
(- b,a)
W
(- 1,4)
(- 4,- 1)
X
(2,3)
(- 3,2)
Y
(1,1)
(- 1,1)
Z
(- 1,2)
(- 2,- 1)
Now we can draw W'X'Y'Z'.
The figures do not have the same orientation just yet, However, given the position of W'X'Y'Z', we can see that if we reflect it in the x-axis, we can map it onto ABCD.
Composition of transformations
Having mapped WXYZ onto ABCD we can now write the compositions of transformations:
Rotation:& 90^(∘) about the origin.
Reflection:& in the x-axis.
b Congruent figures have the same shape and size which means they can be mapped onto each other. Additionally, both rotations and reflections are examples of rigid motions which means angle and sides measures are retained. Since we proved that WXYZ could be mapped onto ABCD using rigid motions, we can be sure that the quadrilaterals are congruent.
