Learning Takeaways: After this lesson, students will be able to understand the effects of:

- translations on coordinates in a Cartesian plane.
- rotations on coordinates in a Cartesian plane.
- reflections on coordinates in a Cartesian plane.

**Important Vocabulary **

**Isometry**: An isometry is a transformation that maintains congruency. This means that the transformation does not change the figure's size or shape.

Make sure your child is familiar with the Cartesian coordinate system including the horizontal x-axis, the vertical y-axis, and the (x,y) convention used for locating points.

If you have not already done so, you may wish to review this congruency lesson with your child.

Translations

Figure J is the pre-image. Figure S is translated.

A figure is a translation if it is moved without rotation.

A *Translation Vector* is a vector that gives the length and direction of a particular translation. Vectors in the Cartesian plane can be written (x,y) which means a translation of *x* units horizontally and *y* units vertically.

Vectors translations can be written as shown in either of these two ways:

This vector can be said to be *ray AB* or* vector D*.

You translate a figure according to the numbers indicated by the vector. So if one point on a figure has coordinates of (-3,3) and the translation vector is (-1,3), the new coordinate is (-4,6). You add or subtract according to the signs in the numbers in the vector.

The red square's coordinates are:

(-4,3); (-4,8); (-9,3); (-9,8).

What are the coordinate pairs under a translation vector of (1,-1) as shown by the blue square?

The new coordinates are:

(-3,2); (-3,7); (-8,2); (-8,7).

You can see that the translation did not move the figure far because the vector translation is small.

We will try a different vector translation of (3,-9) on the same red square

The new coordinates are:

(-1,-6); (-1,-1); (-6,-6); (-6,-1).

Task: Translate ΔABC with a vector translation of (6,-2)

Steps:

- Find the coordinates of Δ ABC.
- Add 6 to the X coordinates and subtract 2 (or add -2) from the Y coordinates.
- Plot the new coordinates on the grid.

Answer and Explanation

Coordinates of ΔABC are:

(-5,1); (-2.5,7); (0,1)

Add 6 and subtract 2

A^{1} = (-5 + 6, 1 - 2) = (1,-1)

B^{1} = (-2.5 + 6, 7 - 2) = (3.5,5)

C^{1} = (0 + 6, 1 - 2) = (6,-1)

Plot ΔA^{1}B^{1}C^{1}

Look at the four triangles on the Cartesian plane below. Each one is rotated about the origin as shown in the table.

Triangle B | Triangle C | Triangle D |

90° rotation of Triangle A about the origin | 90° rotation of Triangle B about the origin | 90° rotation of Triangle C about the origin |

180° rotation of Triangle A about the origin | 180° rotation of Triangle B about the origin | |

| 270° rotation of Triangle A about the origin |

The coordinates are: | |||

Triangle A | (-2,2) | (-10,2) | (-6,8) |

Triangle B | (2,2) | (2,10) | (8,6) |

Triangle C | (2,-2) | (10,-2) | (6,-8) |

Triangle D | (-2,-2) | (-2,-10) | (-8,-6) |

Note the pattern in the coordinates for corresponding vertices on the triangles. |

The coordinates of Pentagon ABCDE are: (0,4); (7,4); (9,2); (7,0); (0,0)

Pentagon ABCDE is rotated 180° about the origin making the coordinates of Pentagon A^{1}B^{1}C^{1}D^{1}E^{1:}

(0,-4); (-7,-4); (-9,-2); (-7,0); (0,0)

Note how the coordinates of corresponding vertices are opposite integers (just the +/- sign is different). This is always the case with 180° rotations about the origin.

The coordinates of LMNO are:

(-7,5); (0,5); (-2,1); (-5,1)

LMNO is reflected over the X-axis making the coordinates of L^{1}M^{1}N^{1}O^{1}:

(-7,-5); (0,-5); (-2,-1); (-5,-1)

Note how the x-coordinates remain the same but the y-coordinates change to their opposite integer (i.e. the sign changes). This is always the case with reflections over the X-axis.

The coordinates of PQSR are:

(-8,-3); (-2,-3); (-2,-6); (-8,-6)

PQSR is reflected over the Y-axis making the coordinates of P^{1}Q^{1}S^{1}R^{1}:

(8,-3); (2,-3); (2,-6); (8,-6)

Note how the y-coordinates remain the same but the x-coordinates change to their opposite integer (i.e. the sign changes). This is always the case with reflections over the Y-axis.