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All courses Geometry · G-CO.4 6 of 57
Define rotations, reflections, and translations using geometric language: angles, circles, perpendicular/parallel lines, and segments.

Define a translation using geometric language

Problem
Describe the translation represented by the vector \(\text{right}~5\), \(\text{up}~2\).
Geometric source diagram presenting the points, line, center, or vector named in the problem. Open full size
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Hint

Read the horizontal and vertical parts of the vector separately: right \(5\), then up \(2\).

A translation moves every point of a figure the same distance and direction, so all image points are connected by parallel directed segments.

Solution walkthrough

01

Read the horizontal component

\[\text{horizontal}~\text{displacement}~=~5~\text{units}~\text{right}\]

The vector's first stated component gives a positive five-unit horizontal movement.

02

Read the vertical component

\[\text{vertical}~\text{displacement}~=~2~\text{units}~\text{up}\]

The second stated component gives a positive two-unit vertical movement.

03

Apply one common vector

\[\text{every}~\text{point}~\text{receives}~\text{displacement}~(5,2)\]

A translation applies the same directed displacement to every point, as the inspected P-to-P-prime arrow illustrates.

04

Describe the translation

\[\text{Every}~\text{point}~\text{moves}~5~\text{units}~\text{right}~\text{and}~2~\text{units}~\text{up}.\]

This sentence includes both vector components and correctly applies them to every point.

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Another way

  1. State that every point follows a parallel directed segment with component displacement (5,2).

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Common mistake

Do not swap the two components. The first describes horizontal movement and the second describes vertical movement.

Annotated geometric diagram showing the defining transformation conditions and result.
definition: every point moves 5 units right and 2 units up along parallel directed segments

Solution walkthrough video