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How to Find the Translation of Figure ABCD to A'B'C'D' on a Coordinate Grid
Mathematics
Grade 8 (Junior High School)
Question Content
Which translation will change figure ABCD to figure A'B'C'D'? The options are: 7 units left and 6 units up; 5 units left and 7 units up; 6 units left and 7 units up; 7 units left and 5 units up. A coordinate grid shows figure ABCD and its translated image A'B'C'D'.
Correct Answer
7 units left and 5 units up
Detailed Solution Steps
1
Step 1: Select a corresponding vertex from the original figure ABCD and its image A'B'C'D', for example, vertex A from ABCD and vertex A' from A'B'C'D'.
2
Step 2: Identify the coordinates of the chosen vertices. Assume vertex A has coordinates (6, -2) and vertex A' has coordinates (-1, 3).
3
Step 3: Calculate the horizontal translation: Subtract the x-coordinate of A from the x-coordinate of A': -1 - 6 = -7. A negative value means a translation to the left by 7 units.
4
Step 4: Calculate the vertical translation: Subtract the y-coordinate of A from the y-coordinate of A': 3 - (-2) = 5. A positive value means a translation upward by 5 units.
5
Step 5: Verify with another pair of corresponding vertices (such as B and B') to confirm the translation is consistent across the entire figure.
Knowledge Points Involved
1
Coordinate Plane Translation
Translation on a coordinate plane is a rigid transformation that shifts every point of a figure the same distance in the same direction. For a point (x, y), translating a units horizontally and b units vertically results in the new point (x+a, y+b), where a is positive for right shifts and negative for left shifts, and b is positive for upward shifts and negative for downward shifts.
2
Corresponding Vertices in Transformations
When a figure is transformed, each point in the original figure maps to exactly one point (its corresponding vertex) in the image. Using corresponding vertices is the standard method to calculate the parameters of a transformation like translation.
3
Rigid Transformations
Rigid transformations (including translations, rotations, and reflections) preserve the size and shape of the original figure. Translations are a type of rigid transformation that only changes the position of the figure, not its orientation, side lengths, or angle measures.
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