Which transformation will result in an image that is congruent to its pre-image?

A. $(x, y) \rightarrow(-x,-4 y)$
B. $(x, y) \rightarrow(4 x, y-4)$
C. $(x, y) \rightarrow(-4 x, y)$
D. $(x, y) \rightarrow(-x,-y)$

Congruent transformations preserve the size and shape of a figure.
Analyze each transformation to see if it preserves congruence.
( x , y ) → ( − x , − 4 y ) , ( x , y ) → ( 4 x , y − 4 ) , and ( x , y ) → ( − 4 x , y ) do not preserve congruence because they involve stretches.
( x , y ) → ( − x , − y ) preserves congruence because it's a 180-degree rotation.
The transformation that results in a congruent image is ( x , y ) → ( − x , − y ) ​ .

Explanation

Understanding Congruence The question asks us to identify which transformation will result in an image that is congruent to its pre-image. A congruent transformation preserves the size and shape of the figure. This means the image and pre-image are identical, just in a different location or orientation.

Analyzing Each Transformation Let's analyze each transformation:



(x, y) -> (-x, -4y): This transformation involves a reflection across the y-axis (due to the -x) and a vertical stretch by a factor of 4 (due to the -4y). The vertical stretch changes the shape, so this transformation does not preserve congruence.
(x, y) -> (4x, y-4): This transformation involves a horizontal stretch by a factor of 4 (due to the 4x) and a vertical translation down by 4 units (due to the y-4). The horizontal stretch changes the shape, so this transformation does not preserve congruence.
(x, y) -> (-4x, y): This transformation involves a reflection across the y-axis (due to the -4x) and a horizontal stretch by a factor of 4 (due to the -4x). The horizontal stretch changes the shape, so this transformation does not preserve congruence.
(x, y) -> (-x, -y): This transformation involves a reflection across the y-axis (due to the -x) and a reflection across the x-axis (due to the -y). Alternatively, this can be seen as a 180-degree rotation about the origin. Rotations preserve size and shape, so this transformation does preserve congruence.


Conclusion Therefore, the transformation that results in an image congruent to its pre-image is ( x , y ) → ( − x , − y ) .

Examples
Imagine you have a photograph. A congruent transformation is like rotating the photograph or flipping it over; the photo still looks the same, just in a different orientation. However, if you stretch the photo, it's no longer congruent to the original because its shape has changed. In architecture, ensuring that scaled models are congruent to the actual building plans is crucial for accurate construction and design. This ensures that all proportions and angles are maintained, guaranteeing the structural integrity and aesthetic appeal of the final building.

Answered by GinnyAnswer | 2025-07-08

The congruent transformation among the options is ( x , y ) → ( − x , − y ) , which is a 180-degree rotation. Other transformations involve stretches or translations, altering the shape and size of the image. Therefore, they do not result in congruence.
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Answered by Anonymous | 2025-07-16