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Geometric Translations

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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.

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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), reflections (discussed elsewhere), and rotations (discussed elsewhere). 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.

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