We will search for the correct option that shows the series of the transformations that maps quadrilateral MNPQ onto quadrilateral RSTU.
To do so we will test each option separately.
Testing Option A
Consider the statement in option A.
Reflection across the x-axis, translation 4 units down.
We will start by reflecting MNPQ across the x-axis.
When we reflect MNPQ across the x-axis, it maps onto quadrilateral RSTU. Therefore, a translation 4 units down after reflection across the x-axis will not map MNPQ onto RSTU, so this option is not correct.
Testing Option B
We will look at the statement in option B.
Reflection across the y-axis, translation 4 units down.
We will start by reflecting MNPQ across the x-axis.
Next, we will translate M'N'P'Q' 4 units down.
Reflection across the y-axis and then a translation 4 units down does not map MNPQ onto RSTU, so this option is not correct.
Rotation 180^(∘) about the origin and then reflection across x-axis.
We will begin by rotating MNPQ. Remember that a rotation is a transformation about a fixed point called the center of rotation. Each point of the original figure and its image are the same distance from the center of rotation. When we perform a counterclockwise rotation about the origin, we can write the coordinates of the image in relation to the coordinates of the preimage.
We want to rotate quadrilateral MNPQ 180^(∘) counterclockwise about the origin. Therefore, we can use the related coordinate changes shown in the table to determine the coordinates of the image of each vertex.
ccc
Preimage & & Image
(x,y) & → & (- x, - y) [0.5em]
M(1,2) & & M'(- 1,- 2) [0.5em]
N(3,3) & & N'(- 3,- 3) [0.5em]
P(2, 4) & & P'(- 2,- 4) [0.5em]
Q(4, 5) & & Q'(- 4,- 5)
Then we will reflect M'N'P'Q' across the x-axis. When we reflect across the x-axis, we multiply the y-values of the coordinates of the preimage's points by - 1. We will begin by reflecting the vertex points of quadrilateral M'N'P'Q'.
ccc
Preimage & & Image
(x,y) & → & ( x,- y) [0.5em]
M'(- 1,- 2) & & M''(- 1, 2) [0.5em]
N'(- 3,- 3) & & N''(- 3, 3) [0.5em]
P'(- 2, - 4) & & P''(- 2, 4) [0.5em]
Q'(- 4, - 5) & & Q''(- 4, 5)
Now we will draw the figure that constructs by the reflection by combining the points. Then we will consider whether the sequence of transformations maps quadrilateral MNPQ onto quadrilateral RSTU.
Since quadrilateral M''N''P''Q'' does not map onto quadrilateral RSTU, option C is an incorrect option.
Testing Option D
Let's find the image of the series of transformations in option D.
Rotation 180^(∘) about the origin and then reflection across y-axis.
We found the image's vertices of quadrilateral MNPQ after a rotation 180^(∘) about the origin for testing option C.
ccc
Preimage & & Image
(x,y) & → & (- x, - y) [0.5em]
M(1,2) & & M'(- 1,- 2) [0.5em]
N(3,3) & & N'(- 3,- 3) [0.5em]
P(2, 4) & & P'(- 2,- 4) [0.5em]
Q(4, 5) & & Q'(- 4,- 5)
Now we will reflect M'N'P'Q' across the y-axis. We will multiply the x-values of the coordinates of preimage's points by - 1. We will begin by reflecting the vertex points of quadrilateral M'N'P'Q'.
ccc
Preimage & & Image
(x,y) & → & (- x, y) [0.5em]
M'(- 1,- 2) & & M''(1,- 2) [0.5em]
N'(- 3,- 3) & & N''(3,- 3) [0.5em]
P'(- 2,- 4) & & P''(2,- 4) [0.5em]
Q'(- 4,- 5) & & Q''(4,- 5)
Now we will draw the image. Then we will consider whether the sequence of transformations maps quadrilateral MNPQ onto quadrilateral RSTU.
Since quadrilateral M''N''P''Q'' overlaps with quadrilateral RSTU, the correct option is D.
