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All courses Geometry · G-CO.5 7 of 57
Draw transformed figures and specify transformation sequences mapping one figure to another.

Translate every vertex of a polygon by the same vector

Problem
Translate the polygon with vertices \(0,~0\); \(3,~0\); \(1,~2\) by the vector \(<4,-1>\).
Coordinate-plane source diagram presenting only the geometric information supplied in the prompt. Open full size
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Hint

Apply the vector <4,-1> to one vertex first: add 4 to x and subtract 1 from y.

A translation by <a,b> sends each vertex (x,y) to (x+a, y+b).

Solution walkthrough

01

Decode the translation vector

\[<4,-1>:~\text{add}~4~\text{to}~x~\text{and}~\text{subtract}~1~\text{from}~y\]

The inspected graph shows source vertices (0,0), (3,0), and (1,2). The vector applies the same coordinate changes to each.

02

Translate the first two vertices

\[(0+4,0-1)=(4,-1);~(3+4,0-1)=(7,-1)\]

Both vertices on the x-axis move four units right and one unit down.

03

Translate the third vertex

\[(1+4,2-1)=(5,1)\]

Applying the same vector to the upper vertex gives x=5 and y=1.

04

List image vertices in source order

\[(4,~-1);~(7,~-1);~(5,~1)\]

These images correspond in order to the prompt's vertices (0,0), (3,0), and (1,2).

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Another way

  1. Move the entire displayed triangle four grid units right and one unit down, then read the new coordinates.

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Common mistake

Do not apply the vector to only one vertex or reverse its signs. Every vertex receives +4 in x and -1 in y.

Annotated coordinate diagram showing the transformed image and defining rule or ordered steps.
image vertices: 4, -1; 7, -1; 5, 1

Solution walkthrough video