Which sequence of transformations could be used to map quadrilateral RSTU onto R"S"T"U"?
A. [tex]R_{0,90^{\circ}} \circ T_{(-3,-1)}[/tex]
B. [tex]T_{(-3,-1)} \circ R_{0,90^{\circ}}[/tex]
C. [tex]R_{0,270^{\circ}} \circ T_{(3,1)}[/tex]
D. [tex]T_{(3,1)} \circ R_{0,270^{\circ}}[/tex]

Translate point (1,1) by (-3,-1) then rotate 90 degrees: (0, -2).
Rotate point (1,1) by 90 degrees then translate by (-3,-1): (-4, 0).
Translate point (1,1) by (3,1) then rotate 270 degrees: (2, -4).
Rotate point (1,1) by 270 degrees then translate by (3,1): (4, 0).

Explanation

Problem Analysis Let's analyze the problem. We need to find the sequence of transformations that maps quadrilateral RSTU onto R"S"T"U". The given options involve rotations and translations. To determine the correct sequence, we can apply each option to a test point and see which one results in a plausible transformation. Let's use the point R(1,1) as our test point.

Applying Option 1 Option 1: R 0 , 9 0 ∘ ​ ∘ T ( − 3 , − 1 ) ​ This means we first translate the point (1,1) by (-3,-1), which gives us (1-3, 1-1) = (-2, 0). Then, we rotate this point 90 degrees counterclockwise about the origin. The rotation rule for 90 degrees counterclockwise is (x, y) -> (-y, x). So, (-2, 0) becomes (0, -2).

Applying Option 2 Option 2: T ( − 3 , − 1 ) ​ ∘ R 0 , 9 0 ∘ ​ This means we first rotate the point (1,1) 90 degrees counterclockwise about the origin, which gives us (-1, 1). Then, we translate this point by (-3,-1), which gives us (-1-3, 1-1) = (-4, 0).

Applying Option 3 Option 3: R 0 , 27 0 ∘ ​ ∘ T ( 3 , 1 ) ​ This means we first translate the point (1,1) by (3,1), which gives us (1+3, 1+1) = (4, 2). Then, we rotate this point 270 degrees counterclockwise about the origin. The rotation rule for 270 degrees counterclockwise is (x, y) -> (y, -x). So, (4, 2) becomes (2, -4).

Applying Option 4 Option 4: T ( 3 , 1 ) ​ ∘ R 0 , 27 0 ∘ ​ This means we first rotate the point (1,1) 270 degrees counterclockwise about the origin, which gives us (1, -1). Then, we translate this point by (3,1), which gives us (1+3, -1+1) = (4, 0).

Summary of Results Without knowing the final coordinates of R" after the transformation, it's impossible to definitively choose the correct option. However, by applying each transformation to the point R(1,1), we have found the resulting coordinates for each option: Option 1: (0, -2) Option 2: (-4, 0) Option 3: (2, -4) Option 4: (4, 0)

Final Answer Since we don't have the final coordinates of R"S"T"U", we cannot determine the correct answer. However, if we assume that Option 4, T ( 3 , 1 ) ​ ∘ R 0 , 27 0 ∘ ​ , is the correct transformation, then the coordinates of R" would be (4,0).


Examples
Transformations are used in computer graphics to manipulate objects on the screen. For example, when you rotate an image or move it across the screen, you are applying transformations to the image's coordinates. These transformations can be combinations of translations, rotations, and scaling, similar to the problem we solved.

Answered by GinnyAnswer | 2025-07-03

To determine which transformation sequence properly maps quadrilateral RSTU to R"S"T"U", each option was evaluated using a test point, producing different resulting coordinates. The four transformation combinations yielded coordinates: (0,-2), (-4,0), (2,-4), and (4,0). Without knowing the final coordinates of R"S"T"U", we cannot definitively choose the correct option but have outlined the transformation results for further evaluation.
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Answered by Anonymous | 2025-07-04