<meta http-equiv="refresh" content="1; url=/nojavascript/"> Geometric Translations ( Read ) | Geometry | CK-12 Foundation
Dismiss
Skip Navigation
You are viewing an older version of this Concept. Go to the latest version.

Geometric Translations

%
Progress
Practice Geometric Translations
Practice
Progress
%
Practice Now
Geometric Translations

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? After completing this Concept, you'll be able to translate a figure like this one in the coordinate plane.

Watch This

Transformation: Translation CK-12

Guidance

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 (discussed here), Transformation: Reflection , and Transformation: Rotation . 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) .

Example A

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.

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

(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)

Example B

Find the translation rule for \triangle TRI to \triangle T'R'I' .

Look at the movement from T to T' . The translation rule is (x,y) \rightarrow (x+6,y-4) .

Example C

Show \triangle TRI \cong \triangle T'R'I' from Example B.

Use the distance formula to find all the lengths of the sides of the two triangles.

& \underline{\triangle TRI} && \underline{\triangle T'R'I'}\\& TR=\sqrt{(-3-2)^2+(3-6)^2}=\sqrt{34} && T'R'=\sqrt{(3-8)^2+(-1-2)^2}=\sqrt{34}\\& RI=\sqrt{(2-(-2))^2+(6-8)^2}=\sqrt{20} && R'I'=\sqrt{(8-4)^2+(2-4)^2}=\sqrt{20}\\& TI=\sqrt{(-3-(-2))^2+(3-8)^2}=\sqrt{26} && T'I'=\sqrt{(3-4)^2+(-1-4)^2}=\sqrt{26}

Since all three pairs of corresponding sides are congruent, the two triangles are congruent by SSS.

Transformation: Translation CK-12

Guided Practice

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

Use the translation (x,y) \rightarrow (x+2,y-5) for questions 2-4.

2. What is the image of A(-6, 3) ?

3. What is the image of B(4, 8) ?

4. What is the image of C(5, -3) ?

Answers:

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

& 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)

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

2. A' (-4, -2)

3. B' (6, 3)

4. C' (7, -8)

Practice

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

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

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

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

In questions 14-17, \triangle A'B'C' is the image of \triangle ABC . Write the translation rule.

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

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

Vocabulary

rigid transformation

rigid transformation

A transformation that does not change the size or shape of a figure, also known as an isometry or congruence transformation.
transformation

transformation

An operation that moves, flips, or otherwise changes a figure to create a new figure. The new figure created by a transformation is called the image. The original figure is called the preimage.
translation

translation

A transformation that moves every point in a figure the same distance in the same direction.

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Geometric Translations.

Reviews

Please wait...
Please wait...

Original text