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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

Vocabulary

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 new figure created by a transformation is called the image. The original figure is called the preimage. A translation is a transformation that moves every point in a figure the same distance in the same direction. A vector is a quantity that has direction and size. The component form of a vector combines the horizontal distance traveled and the vertical distance traveled.

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).

Practice

  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'.

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