# 8.5: Geometric Translation

- Page ID
- 2169

Understand translations as movement of every point in a figure the same distance in the same direction. Graph images given preimage and translation.

## Translations

A **transformation** is an operation that moves, flips, or otherwise changes a figure to create a new figure. A **rigid transformation** (also known as an **isometry** or **congruence transformation**) is a transformation that does not change the size or shape of a figure.

The rigid transformations are **translations**, reflections, and rotations. The new figure created by a transformation is called the **image**. The original figure is called the **preimage**. If the preimage is \(A\), then the image would be \(A′\), said “a prime.” If there is an image of \(A′\), that would be labeled \(A′′\), said “a double prime.”

A **translation** is a transformation that moves every point in a figure the same distance in the same direction. For example, this transformation moves the parallelogram to the right 5 units and up 3 units. It is written \((x,y)\rightarrow (x+5, y+3)\).

What if you were given the coordinates of a quadrilateral and you were asked to move that quadrilateral 3 units to the left and 2 units down? What would its new coordinates be?

Example \(\PageIndex{1}\)

Triangle \(\Delta ABC\) has coordinates \(A(3,−1)\), \(B(7,−5)\) and \(C(−2,−2)\). Translate \(\Delta ABC\) to the left 4 units and up 5 units. Determine the coordinates of \(\Delta A′B′C′\).

**Solution**

Graph \(\Delta ABC\). To translate \(\Delta ABC\), subtract 4 from each \(x\) value and add 5 to each \(y\) value of its coordinates.

\(\begin{aligned} &A(3,−1)\rightarrow (3−4,−1+5)=A′(−1,4) \\ &B(7,−5)\rightarrow (7−4,−5+5)=B′(3,0) \\ &C(−2,−2)\rightarrow (−2−4,−2+5)=C′(−6,3) \end{aligned}\)

The rule would be \((x,y)\rightarrow (x−4, y+5)\).

Example \(\PageIndex{2}\)

Using the translation \((x,y)\rightarrow (x+2, y−5)\), what is the image of \(A(−6, 3)\)?

**Solution**

\(A′(−4,−2)\)

Example \(\PageIndex{3}\)

Graph square \(S(1,2)\), \(Q(4,1)\), \(R(5,4)\) and \(E(2,5)\). Find the image after the translation \((x,y)\rightarrow (x−2,y+3)\). Then, graph and label the image.

**Solution**

We are going to move the square to the left 2 and up 3.

\(\begin{aligned}(x,y)&\rightarrow (x−2,y+3) \\ S(1,2)&\rightarrow S′(−1,5) \\ Q(4,1)&\rightarrow Q′(2,4) \\ R(5,4)&\rightarrow R′(3,7) \\ E(2,5)&\rightarrow E′(0,8)\end{aligned}\)

Example \(\PageIndex{4}\)

Find the translation rule for \(\Delta TRI\) to \(\Delta T′R′I′\).

**Solution**

Look at the movement from \(T\) to \(T′\). The translation rule is \((x,y)\rightarrow (x+6, y−4)\).

## Review

Use the translation \((x,y)\rightarrow (x+5, y−9)\) for questions 1-7.

- What is the image of \(A(−1,3)\)?
- What is the image of \(B(2,5)\)?
- What is the image of \(C(4,−2)\)?
- What is the image of \(A′\)?
- What is the preimage of \(D′(12,7)\)?
- What is the image of \(A′′\)?
- Plot \(A\), \(A′\), \(A′′\), and \(A′′′\) from the questions above. What do you notice?

The vertices of \(\Delta ABC\) are \(A(−6,−7)\), \(B(−3,−10)\) and \(C(−5,2)\). Find the vertices of \(\Delta A′B′C′\), given the translation rules below.

- \((x,y)\rightarrow (x−2, y−7)\)
- \((x,y)\rightarrow (x+11, y+4)\)
- \((x,y)\rightarrow (x, y−3)\)
- \((x,y)\rightarrow (x−5, y+8)\)
- \((x,y)\rightarrow (x+1, y)\)
- \((x,y)\rightarrow (x+3, y+10)\)

In questions 14-17, \(\Delta A′B′C′\) is the image of \(\Delta ABC\). Write the translation rule.

Use the triangles from #17 to answer questions 18-20.

- Find the lengths of all the sides of \(\Delta ABC\).
- Find the lengths of all the sides of \(\Delta A′B′C′\).
- What can you say about \(\Delta ABC\) and \(\Delta A′B′C′\)? Can you say this for
translation?__any__ - If \(\Delta A′B′C′\) was the
and \(\Delta ABC\) was the image, write the translation rule for #14.__preimage__ - If \(\Delta A′B′C′\) was the
and \(\Delta ABC\) was the image, write the translation rule for #15.__preimage__ - Find the translation rule that would move \(A\) to \(A′(0,0)\), for #16.
- The coordinates of \(\Delta DEF\) are \(D(4,−2)\), \(E(7,−4)\) and \(F(5,3)\). Translate \(\Delta DEF\) to the right 5 units and up 11 units. Write the translation rule.
- The coordinates of quadrilateral \(QUAD\) are \(Q(−6,1)\), \(U(−3,7)\), \(A(4,−2)\) and \(D(1,−8)\). Translate \(QUAD\) to the left 3 units and down 7 units. Write the translation rule.

## Review (Answers)

To see the Review answers, open this PDF file and look for section 12.3.

## Additional Resources

Interactive Element

Video: Transformation: Translation Principles - Basic

Activities: Translations Discussion Questions

Study Aids: Types of Transformations Study Guide

Practice: Geometric Translation