Most high school geometry students can tell you that one of the most challenging aspects of the class is the emphasis on the nature of how points or shapes in a two-dimensional plane are transformed. A plane describes a two-dimensional surface, such as a traditional cartesian graph or the shapes drawn into it. Shapes aside, even equations graphed into the plane can be transformed through one of four methods: reflection, rotation, translation, and dilation. This is also known as flip, turn, slide, and resize. Today we will be focusing on the geometry transformation known as translation.

### What are Geometry Translations?

In geometry, a translation describes a transformation that does not change the shape or direction of the objects on the graph, but only their location on the graph. This means the figure and its translation will always be congruent since the size will never be impacted. In other words, the overall geometry of the figure will not change in appearance in either type of transformation that occurs. The two types of transformations in plane geometry are vertical and horizontal.

### Vertical Translations** **

Coordinates for ABC | Coordinates for ABC’ |

A: (x_{1},y_{1}) |
A’:(x_{1},(y_{1}+8)) |

B: (x_{2},y_{2}) |
B’: (x_{2},(y_{2}+8)) |

C: (x_{3},y_{3}) |
C’: (x_{3},(y_{3}+8)) |

As seen in the graph and table above, a vertical translation refers to modifying the value of the *y*-coordinate by a constant value, allowing the plane figure to shift up or down the graph along the *y*-axis. Positive values added to the *y *coordinate will shift the shape upwards and negative values will shift the shape downwards. The *x* value does not change unless there is also a horizontal transformation as well.

** **

### Horizontal Translations** **

Coordinates for ABC | Coordinates for ABC’ |

A: (x_{1},y_{1}) |
A’:((x_{1}+8),y_{1}) |

B: (x_{2},y_{2}) |
B’: ((x_{2}+8),y_{2}) |

C: (x_{3},y_{3}) |
C’: ((x_{3}+8),y_{3}) |

** **

In this graph, we are only viewing a horizontal translation, which impacts only the *x*-axis. The horizontal shift to the right will be represented by the expression *x *+ *k*. The horizontal shift to left will be expressed as *x *– *k*. The horizontal shift exists separately from the vertical shift.

### Practical Applications for Translations Beyond Geometry

While translations are familiar to students taking geometry classes in high school, the lessons of translation are easily transferable to other fields. They can be used in physics for understanding kinematics and vectors, as well as fields far away from the graphs.

Physics: Understanding the distance an object translates is helpful for understanding kinematics and Newton’s Laws. Knowing the direction, distance, and acceleration of an object is all useful information for calculating forces and time. Another area where both physics and translations will interact with each other lies in roller coasters. The car in the roller coaster will translate within its track both vertically and horizontally, but the car and its occupants will not change in size either.

Vectors: Translations are a form of adding or subtracting a constant vector of a shape. Vectors are critical for understanding navigation by boat or air because the speed of the vessel and of other factors like the wind or river impact the direction the object will translate towards. Often even translating a few degrees off in direction will make a difference of thousands of miles where airplanes are concerned.

Sports: When considering batting sports like baseball, softball, or cricket, a translation does occur when the ball makes contact with the hitting surface. When the ball goes flying, both the horizontal and vertical translations contribute to its distance, and where it will eventually land will determine which team earns the points and either wins or loses the game.

There are certain steps to understanding how translations in geometry occur, which determine how a shape will shift with a graph. However, it is also important to recognize that there are ways to apply the concept of translations beyond the graph.

Author: Maaida Kirmani