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12.2: Translations

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

  • Graph a point, line, or figure and translate it x and y units.
  • Write a translation rule.

Review Queue

  1. Find the equation of the line that contains (9, -1) and (5, 7).
  2. What type of quadrilateral is formed by A(1, -1), B(3, 0), C(5, -5) and D(-3, 0)?
  3. Find the equation of the line parallel to #1 that passes through (4, -3).

Know What? The distances between San Francisco, S, Paso Robles, P, and Ukiah, U, are given in miles the graph. Find:

a) The translation rule for P to S.

b) The translation rule for S to U.

c) The translation rule for P to U.

d) The translation rule for U to S. It is not the same as part b.


Transformation: An operation that moves, flips, or changes a figure to create a new figure.

Rigid Transformation: 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. Another word for a rigid transformation is an isometry or congruence transformations.

In Lesson 7.6, we learned how to label an image. 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.”


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

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 1: 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.

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

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

Example 3: Show \triangle TRI \cong \triangle T'R'I' from Example 2.

Solution: 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}

This verifies our statement at the beginning of the section that a translation is an isometry or congruence translation.

Example 4: 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'.

Solution: Graph \triangle ABC. To translate \triangle ABC, subtract 4 from each x value and add 5 to each y value.

& 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).

Know What? Revisited

a) (x,y) \rightarrow (x-84,y+187)

b) (x,y) \rightarrow (x-39,y+108)

c) (x,y) \rightarrow (x-123,y+295)

d) (x,y) \rightarrow (x+39,y-108)

Review Questions

  • Questions 1-13 are similar to Example 1.
  • Questions 14-17 are similar to Example 2.
  • Questions 18-20 are similar to Example 3.
  • Questions 21-23 are similar to Example 1.
  • Questions 24 and 25 are similar to Example 4.

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

  1. What is the image of A(-6, 3)?
  2. What is the image of B(4, 8)?
  3. What is the image of C(5, -3)?
  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.

Review Queue Answers

  1. y = -2x+17
  2. Kite
  3. y = -2x+5

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