Texas Essential Knowledge and Skills (TEKS):
8.10.A

generalize the properties of orientation and congruence of rotations, reflections, translations, and dilations of two-dimensional shapes on a coordinate plane;

Texas Essential Knowledge and Skills (TEKS):
8.10.B

differentiate between transformations that preserve congruence and those that do not;

Texas Essential Knowledge and Skills (TEKS):
8.10.C

explain the effect of translations, reflections over the x- or y-axis, and rotations limited to 90°, 180°, 270°, and 360° as applied to two-dimensional shapes on a coordinate plane using an algebraic representation; and

8th Grade Math - Transformations Lesson

A translation is a transformation in which a figure's position changes. Its size, shape, and orientation do not change.

Given a point (x, y), a translation of a units to the right affects the x-coordinate in the positive direction. So, if the point (x, y) is translated a units to the right, the new coordinates are (x + a, y).

Similarly, if the point (x, y) is translated a units to the left, the new coordinates are (x - a, y).

Given a point (x, y), a translation of b units up affects the y-coordinate in the positive direction. So, if the point (x, y) is translated b units up, the new coordinates are (x, y + b).

Similarly, if the point (x, y) is translated b units down, the new coordinates are (x, y - b).

Example:

Translating to the right means the x-coordinate increases. Translating down means the y-coordinate decreases.

So, if the vertex which was located at (-2, 7) is translated 3 units to the right and 2 units down, then the new coordinates are (-2 + 3, 7 - 2), or (1, 5).

A reflection is a transformation in which a figure's position and orientation change. Its size and shape do not change.

Given a point (x, y), a reflection across the x-axis affects the y-coordinate. The reflection affects which quadrant the point is in, taking it from a negative y-coordinate to positive, or a positive y-coordinate to negative. So, if the point (x, y) is reflected across the x-axis, the new coordinates are (x, -y). Similarly, if the point (x, y) is reflected across the y-axis, the new coordinates are (-x, y).

Example:

A reflection across the x-axis affects the sign of the y-coordinate. So, if the vertex which was located at (-1, 1) is reflected across the x-axis, then the new coordinates are (-1, -1).

A reflection across the y-axis affects the sign of the x-coordinate. So, if the vertex which was located at (-1, 1) is reflected across the y-axis, then the new coordinates are (1, 1).

A rotation is a transformation in which a figure's position changes. Its size, shape, and orientation do not change.

If the point (x, y) is rotated 90° counterclockwise about the origin, the new coordinates are (-y, x).

If the point (x, y) is rotated 180° counterclockwise about the origin, the new coordinates are (-x, -y).

If the point (x, y) is rotated 270° counterclockwise about the origin, the new coordinates are (y, -x).

In the image below, triangle GHI is located in quadrant 4. Quadrant 1 shows triangle GHI rotated 90° counterclockwise about the origin. Quadrant 2 shows triangle GHI rotated 180° counterclockwise about the origin. Quadrant 3 shows triangle GHI rotated 270° counterclockwise about the origin.

A dilation is a transformation in which a figure's size changes. The image which results from a dilation is similar to the original image.

The reference point by which an image is dilated is referred to as the center of dilation. For this lesson, the center of dilation is the origin.

Since a dilation results in an image which is similar to the original image, the images differ by a scale factor. If the scale factor is greater than 1, the new image is larger than the original image. This image is often referred to as an enlargement. If the scale factor is between 0 and 1, the new image is smaller than the original image. This image is often referred to as a reduction.

If a figure in the plane is dilated through the origin by a scale factor of k, then for any (x, y) of that figure, the new coordinates are (kx, ky).

Example:

The table shows how the dilation affects each vertex of the figure.

Original Image Coordinates

Apply Scale Factor

New Image Coordinates

(-4, 0)

(-4 • 2, 0 • 2)

(-8, 0)

(-4, -3)

(-4 • 2, -3 • 2)

(-8, -6)

(3, 0)

(3 • 2, 0 • 2)

(6, 0)

(3, -3)

(3 • 2, -3 • 2)

(6, -6)

How Transformations Affect Similarity and Congruence

Rotations, reflections, and translations do not affect a figure's size or shape. So, any series of rotations, reflections, and/or translations will result in a figure which is congruent to the original figure.

Dilations affect a figure's size, but not its shape. So, any series of dilations, rotations, reflections, and/or translations will result in a figure which is similar, but not necessarily congruent, to the original figure.