Question

the rule \\(r_{y=x} \circ t_{4, 0}(x, y)\\) is applied to trapezoid abcd to produce the final image a\b\c\d\.

which ordered pairs name the coordinates of vertices of the pre-image, trapezoid abcd? select two options.

\\(\square\\ (-1, 0)\\)
\\(\square\\ (-1, -5)\\)
\\(\square\\ (1, 1)\\)
\\(\square\\ (7, 0)\\)
\\(\square\\ (7, -5)\\)

Explanation:

Identify coordinates of the final image

Identify the coordinates of the vertices of the final image trapezoid \(A''B''C''D''\) from the given coordinate grid:

$$
LATEXBLOCK0
$$

Apply the inverse reflection

Using the Geometric Reflections and Composition of Transformations knowledge points

$$
LATEXBLOCK1
$$

Apply the inverse translation

Find the pre-image coordinates by applying the inverse translation \(T_{4, 0}^{-1} = T_{-4, 0}\):

$$
LATEXBLOCK2
$$

Match with the given options

Compare the calculated pre-image vertices \(\{(-1, 0), (-1, -5), (1, -1), (1, -4)\}\) with the options:

• \((-1, 0)\) is a vertex (matches \(C\)).
• \((-1, -5)\) is a vertex (matches \(D\)).
• \((1, 1)\) is not a vertex.
• \((7, 0)\) is not a vertex.
• \((7, -5)\) is not a vertex.

Answer:

• (-1, 0) (Correct answer)
• (-1, -5) (Correct answer)
• (1, 1)
• (7, 0)
• (7, -5)