Rectangle $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is the image of rectangle $A B C D$ after it has been translated according to the rule $T_{-4,3}(x, y)$.
Which points are vertices of the pre-image, rectangle ABCD? Select four options.
$(-1,-2)$
$(7,1)$
$(-1,7)$
$(-1,1)$
$(7,-2)$
Answer (2)
Apply the inverse translation T 4 , − 3 ( x , y ) to each of the given points.
Calculate the new coordinates: ( 3 , − 5 ) , ( 11 , − 2 ) , ( 3 , 4 ) , ( 3 , − 2 ) , and ( 11 , − 5 ) .
Identify the four points that can form a rectangle: ( 3 , − 5 ) , ( 11 , − 5 ) , ( 3 , − 2 ) , and ( 11 , − 2 ) .
Determine the original points corresponding to these pre-image vertices: ( − 1 , − 2 ) , ( 7 , 1 ) , ( − 1 , 1 ) , ( 7 , − 2 ) .
Explanation
Analyze the problem and given data The problem states that rectangle A ′ B ′ C ′ D ′ is the image of rectangle A BC D after a translation T − 4 , 3 ( x , y ) . This means that each point ( x , y ) in rectangle A BC D is mapped to ( x − 4 , y + 3 ) in rectangle A ′ B ′ C ′ D ′ . We are given five points, which are potential vertices of the image rectangle A ′ B ′ C ′ D ′ , and we need to find which four of them are the vertices of the pre-image rectangle A BC D .
Determine the inverse translation To find the vertices of the pre-image rectangle A BC D , we need to apply the inverse translation to the given points. The inverse translation of T − 4 , 3 ( x , y ) is T 4 , − 3 ( x , y ) , which maps a point ( x , y ) to ( x + 4 , y − 3 ) . We will apply this inverse translation to each of the five given points.
Apply the inverse translation to each point Let's apply the inverse translation T 4 , − 3 ( x , y ) to each of the given points:
( − 1 , − 2 ) → ( − 1 + 4 , − 2 − 3 ) = ( 3 , − 5 )
( 7 , 1 ) → ( 7 + 4 , 1 − 3 ) = ( 11 , − 2 )
( − 1 , 7 ) → ( − 1 + 4 , 7 − 3 ) = ( 3 , 4 )
( − 1 , 1 ) → ( − 1 + 4 , 1 − 3 ) = ( 3 , − 2 )
( 7 , − 2 ) → ( 7 + 4 , − 2 − 3 ) = ( 11 , − 5 )
Identify the vertices of the pre-image rectangle Now we have the following points as potential vertices of the pre-image rectangle A BC D : ( 3 , − 5 ) , ( 11 , − 2 ) , ( 3 , 4 ) , ( 3 , − 2 ) , and ( 11 , − 5 ) .
We need to select four points that can form a rectangle. Notice that ( 3 , − 5 ) and ( 11 , − 5 ) have the same y-coordinate, so they lie on the same horizontal line. Also, ( 3 , − 2 ) and ( 11 , − 2 ) have the same y-coordinate, so they lie on the same horizontal line. Furthermore, ( 3 , − 5 ) and ( 3 , − 2 ) have the same x-coordinate, so they lie on the same vertical line. Similarly, ( 11 , − 5 ) and ( 11 , − 2 ) have the same x-coordinate, so they lie on the same vertical line.
Therefore, the four points that form a rectangle are ( 3 , − 5 ) , ( 11 , − 5 ) , ( 3 , − 2 ) , and ( 11 , − 2 ) .
These points correspond to the pre-image of the following points:
( 3 , − 5 ) is the pre-image of ( − 1 , − 2 )
( 11 , − 2 ) is the pre-image of ( 7 , 1 )
( 3 , 4 ) is not a vertex of the pre-image rectangle
( 3 , − 2 ) is the pre-image of ( − 1 , 1 )
( 11 , − 5 ) is the pre-image of ( 7 , − 2 )
State the final answer Thus, the vertices of the pre-image rectangle A BC D are the pre-images of the points ( − 1 , − 2 ) , ( 7 , 1 ) , ( − 1 , 1 ) , and ( 7 , − 2 ) .
Examples
Translations are used in computer graphics to move objects around the screen. For example, if you have a character in a video game and you want to move it 5 units to the right and 3 units up, you would apply the translation T 5 , 3 ( x , y ) to the coordinates of each vertex of the character. This ensures that the character moves as a whole without changing its shape or size.
The vertices of the pre-image rectangle ABCD corresponding to the translated rectangle A'B'C'D' are ( − 1 , − 2 ) , ( 7 , 1 ) , ( − 1 , 1 ) , and ( 7 , − 2 ) . The inverse translation was applied to the given points to find these original points. Hence, the correct answer is the four selected options as pre-image vertices.
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