Write the translation matrix for each figure. Then find the...

Write the translation matrix for each figure. Then find the coordinates of the image after the translation. Graph the preimage and the image on a coordinate plane. Rectangle RSUT with vertices $R(-3,2), S(1,2), U(1,-1), T(-3,-1)$ is translated so that $T^{\prime}$ is at $(-4,1)$ . Find the coordinates of $R^{\prime}$ and $U^{\prime}$ .

Write the translation matrix for each figure. Then find the coordinates of the image after the translation. Graph the preimage and the image on a coordinate plane.
Rectangle RSUT with vertices $R(-3,2), S(1,2), U(1,-1), T(-3,-1)$ is translated so that $T^{\prime}$ is at $(-4,1)$ . Find the coordinates of $R^{\prime}$ and $U^{\prime}$ .

Key Concepts

Graphing in the Coordinate Plane

Graphing involves plotting points and figures on the coordinate plane to visually represent their positions before and after a transformation. In the context of translations, graphing both the original figure (preimage) and its shifted version (image) helps in understanding the effect of the translation and verifying the correctness of the transformation.

Vector Addition

Translations are implemented through vector addition where the translation vector, composed of horizontal and vertical components, is added to each point’s coordinate. This operation shifts every point by the same amount, making it a fundamental concept in understanding how points and figures move within the coordinate plane.

Translation Transformation

A translation is a type of geometric transformation that moves every point of a shape or object the same distance in a specified direction. It is a rigid motion, meaning that the shape and size remain unchanged, and involves shifting the entire figure from one position to another without any rotation or reflection.

Translation Matrix

In coordinate geometry, a translation can be represented by a matrix when using homogeneous coordinates. The translation matrix typically takes the form of a 3x3 matrix, such as [[1, 0, a], [0, 1, b], [0, 0, 1]], where ‘a’ and ‘b’ represent the horizontal and vertical shifts, respectively. This matrix allows the translation to be performed using matrix multiplication.

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