9-1 Translations

Quick Review

A transformation of a geometric figure is a change in its position, shape, or size. An isometry is a transformation in which the preimage and the image are congruent.

A translation is an isometry that maps all points of a figure the same distance in the same direction.

In a composition of transformations, each transformation is performed on the image of the preceding transformation.

Example

What are the coordinates of the image of A ( 5 , − 9 ) eh open 5 comma negative 9 close  for the translation ( x , y ) → ( x − 2 , y + 3 ) ? open bold italic x comma bold italic y close rightwards arrow open bold italic x minus 2 comma bold italic y plus 3 close question mark

Substitute 5 for x and − 9 negative 9  for y in the rule.

A ( 5 , 9 ) → ( 5 − 2 , − 9 + 3 ) , or  A ′ ( 3 , − 6 ) . eh open 5 comma 9 close rightwards arrow open 5 minus 2 comma negative 9 plus 3 close comma or , eh prime , open 3 comma negative 6 close .

Exercises

5. 1. A transformation maps ZOWE onto LFMA. Does the transformation appear to be an isometry? Explain.

   2. What is the image of Z E ¯ ? z e bar , question mark  What is the preimage of M?

      

6. Δ R S T cap delta r s t  has vertices R ( 0 , − 4 ) , S ( − 2 , − 1 ) , r open 0 comma negative 4 close comma s open negative 2 comma negative 1 close comma  and T ( − 6 , 1 ) . t open negative 6 comma 1 close .  Graph the image of Δ R S T cap delta r s t  for the translation ( x , y ) → ( x − 4 , y + 7 ) . open x comma y close rightwards arrow open x minus 4 comma y plus 7 close .
7. Write a rule to describe a translation 5 units left and 10 units up.
8. Find a single translation that has the same effect as the following composition of translations. ( x , y ) → ( x − 5 , y + 7 ) open x comma y close rightwards arrow open x minus 5 comma y plus 7 close  followed by ( x , y ) → ( x + 3 , y ) open x comma y close rightwards arrow open x plus 3 comma y close

9-2 and 9-3 Reflections and Rotations

Quick Review

The diagram shows a reflection across line r. A reflection is an isometry in which a figure and its image have opposite orientations.

The diagram shows a rotation of x ° x degrees  about point R. A rotation is an isometry in which a figure and its image have the same orientation.

Example

Use points P ( 1 , 0 ) , Q ( 3 , − 2 ) , and  R ( 4 , 0 ) . p open 1 comma 0 close comma q open 3 comma negative 2 close comma , and r open 4 comma 0 close .  What is the image of Δ P Q R cap delta p q r  reflected across the y-axis?

Graph Δ P Q R . cap delta p q r .  Find P ′ , Q ′ , and  R ′ p prime , comma , q prime , comma , and , r prime  such that the y-axis is the perpendicular bisector of P P ′ ¯ , Q Q ′ ¯ , p , p prime bar . comma . q , q prime bar . comma  and R R ′ ¯ . r , r prime bar . .  Draw Δ P ′ Q ′ R ′ . cap delta , p prime , q prime , r prime , .

Exercises

Given points A ( 6 , 4 ) , B ( − 2 , 1 ) , and  C ( 5 , 0 ) , eh open 6 comma 4 close comma b open negative 2 comma 1 close comma , and c open 5 comma 0 close comma  graph Δ A B C cap delta eh b c  and its reflection image across each line.

9. the x-axis
10. x = 4 x equals 4
11. y = x y equals x

12. Copy the diagram. Then draw the image of Δ Z X Y cap delta z x y  for a 90 ° 90 degrees  rotation about P. Label the vertices of the image using prime notation.

    

13. Find the image of P ( − 4 , 1 ) p open negative 4 comma 1 close  for a 180 ° 180 degrees  rotation about the origin.

Point O is the center of regular pentagon NMPQR.

14. What is the image of point N for a composition of a 72 ° 72 degrees  rotation and a 144 ° 144 degrees  rotation about O?

    

15. What is the angle of rotation that maps point P to point Q?

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