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Geometric Translations

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Geometric Translations

What if Lucy lived in San Francisco, S , and her parents lived in Paso Robles, P ? She will be moving to Ukiah, U , in a few weeks. All measurements are in miles. Find:

a) The component form of \stackrel{\rightharpoonup}{PS}, \stackrel{\rightharpoonup}{SU} and \stackrel{\rightharpoonup}{PU} .

b) Lucy’s parents are considering moving to Fresno, F . Find the component form of \stackrel{\rightharpoonup}{PF} and \stackrel{\rightharpoonup}{UF} .

c) Is Ukiah or Paso Robles closer to Fresno?

After completing this Concept, you'll be able to answer these questions.

Watch This

CK-12 Foundation: Chapter12TranslationsA

Learn more about translations by watching the video at this link.

Guidance

A transformation is an operation that moves, flips, or changes a figure to create a new figure. A rigid transformation is a transformation that preserves size and shape. The rigid transformations are: translations (discussed here), 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 . Rigid transformations are also called congruence transformations . If the preimage is A , then the image would be labeled 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. In the coordinate plane, we say that a translation moves a figure x units and y units. Another way to write a translation rule is to use vectors. A vector is a quantity that has direction and size.

In the graph below, the line from A to B , or the distance traveled, is the vector. This vector would be labeled \stackrel{\rightharpoonup}{AB} because A is the initial point and B is the terminal point . The terminal point always has the arrow pointing towards it and has the half-arrow over it in the label.

The component form of \stackrel{\rightharpoonup}{AB} combines the horizontal distance traveled and the vertical distance traveled. We write the component form of \stackrel{\rightharpoonup}{AB} as \left \langle 3, 7 \right \rangle because \stackrel{\rightharpoonup}{AB} travels 3 units to the right and 7 units up. Notice the brackets are pointed, \left \langle 3, 7 \right \rangle , not curved.

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.

The translation notation tells us that 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

Name the vector and write its component form.

The vector is \stackrel{\rightharpoonup}{DC} . From the initial point D to terminal point C , you would move 6 units to the left and 4 units up. The component form of \stackrel{\rightharpoonup}{DC} is \left \langle -6, 4 \right \rangle .

Example C

Name the vector and write its component form.

The vector is \stackrel{\rightharpoonup}{EF} . The component form of \stackrel{\rightharpoonup}{EF} is \left \langle 4, 1 \right \rangle .

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter12TranslationsB

Concept Problem Revisited

a) \stackrel{\rightharpoonup}{PS}= \left \langle -84, 187 \right \rangle, \stackrel{\rightharpoonup}{SU} = \left \langle -39, 108 \right \rangle, \stackrel{\rightharpoonup}{PU} = \left \langle -123, 295 \right \rangle

b) \stackrel{\rightharpoonup}{PF} = \left \langle 62, 91 \right \rangle,\stackrel{\rightharpoonup}{UF} = \left \langle 185, -204 \right \rangle

c) You can plug the vector components into the Pythagorean Theorem to find the distances. Paso Robles is closer to Fresno than Ukiah.

UF = \sqrt{185^2 + (-204)^2} \cong 275.4 \ miles, PF = \sqrt{62^2 + 91^2} \cong 110.1 \ miles

Guided Practice

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

2. Draw the vector \stackrel{\rightharpoonup}{ST} with component form \left \langle 2, -5 \right \rangle .

3. Triangle \triangle ABC has coordinates A(3, -1), B(7, -5) and C(-2, -2) . Translate \triangle ABC using the vector \left \langle -4, 5 \right \rangle . Determine the coordinates of \triangle A'B'C' .

4. Write the translation rule for the vector translation from #3.

Answers:

1. Look at the movement from T to T' . T is (-3, 3) and T' is (3, -1). The change in x is 6 units to the right and the change in y is 4 units down. Therefore, the translation rule is (x,y) \rightarrow (x + 6, y - 4) .

2. The graph is the vector \stackrel{\rightharpoonup}{ST} . From the initial point S it moves down 5 units and to the right 2 units.

3. It would be helpful to graph \triangle ABC . To translate \triangle ABC , add each component of the vector to each point to find \triangle A'B'C' .

A(3, -1) + \left \langle -4, 5 \right \rangle & = A'(-1, 4)\\B(7, -5) + \left \langle -4, 5 \right \rangle & = B'(3,0)\\C(-2, -2) + \left \langle -4, 5 \right \rangle & = C'(-6, 3)

4. To write \left \langle -4, 5 \right \rangle as a translation rule, it would be (x, y) \rightarrow (x - 4, y + 5) .

Explore More

  1. What is the difference between a vector and a ray?

Use the translation (x, y) \rightarrow (x + 5, y - 9) for questions 2-8.

  1. What is the image of A(-6, 3) ?
  2. What is the image of B(4, 8) ?
  3. What is the preimage 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? Write a conjecture.

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)

In questions 13-16, \triangle A'B'C' is the image of \triangle ABC . Write the translation rule.

For questions 17-19, name each vector and find its component form.

  1. The coordinates of \triangle DEF are D(4, -2), E(7, -4) and F(5, 3) . Translate \triangle DEF using the vector \left \langle 5, 11 \right \rangle and find the coordinates of \triangle D'E'F' .
  2. The coordinates of quadrilateral QUAD are Q(-6, 1), U(-3, 7), A(4, -2) and D(1, -8) . Translate QUAD using the vector \left \langle -3, -7 \right \rangle and find the coordinates of Q'U'A'D' .

Vocabulary

Center of Rotation

Center of Rotation

In a rotation, the center of rotation is the point that does not move. The rest of the plane rotates around this fixed point.
Image

Image

The image is the final appearance of a figure after a transformation operation.
Preimage

Preimage

The pre-image is the original appearance of a figure in a transformation operation.
Transformation

Transformation

A transformation moves a figure in some way on the coordinate plane.
Translation

Translation

A translation is a transformation that slides a figure on the coordinate plane without changing its shape, size, or orientation.

Image Attributions

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