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. Triangle $D E F$ with vertices $D(-2,2), E(3,5),$ and $F(5,-2)$ is translated so that $D^{\prime}$ is at $(1,-5) .$ Find the coordinates of $E^{\prime}$ and $F^{\prime} .$

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.
Triangle $D E F$ with vertices $D(-2,2), E(3,5),$ and $F(5,-2)$ is translated so that $D^{\prime}$ is at $(1,-5) .$ Find the coordinates of $E^{\prime}$ and $F^{\prime} .$

Key Concepts

Graphing on a Coordinate Plane

Graphing in the coordinate plane involves plotting the preimage and its translated image by marking their vertices and connecting them appropriately. This visual representation helps to illustrate the effects of the translation transformation by showing the relative positions of the original and moved figures.

Preimage and Image

The preimage refers to the original figure before a transformation is applied, while the image is the resulting figure after the transformation. In the context of a translation, each vertex of the preimage is shifted by the same vector to produce the corresponding vertex in the image.

Translation

A translation is a type of geometric transformation that moves every point of a figure or shape by the same distance in a given direction. This transformation does not change the size, shape, or orientation of the figure, making it a rigid motion.

Translation Matrix

A translation matrix is a tool used in the representation of translation operations in a coordinate system, often expressed in augmented or homogeneous coordinates. In two dimensions, the translation can be incorporated into a 3x3 matrix that, when multiplied by a coordinate vector expressed in homogeneous form, shifts the coordinates by the translation vector.

Vector Addition

Translation in the coordinate plane is achieved by adding a constant translation vector to every point of the original figure. This process involves adding the horizontal and vertical components of the vector to the x- and y-coordinates of each vertex, respectively.

Transcript

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