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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.

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.

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.

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 $Q'U'A'D'$.