Take Note Key Concept: Translation

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

A translation is an isometry.

A A ′ = B B ′ = C C ′ eh eh prime equals b b prime equals c c prime

Problem 3 Finding the Image of a Translation

What are the images of the vertices of Δ P Q R cap delta p q r  for the translation ( x , y ) → ( x − 2 , y − 5 ) ? open bold italic x comma bold italic y close rightwards arrow open bold italic x minus 2 comma bold italic y minus 5 close question mark  Graph the image of Δ P Q R . cap delta p q r .

Identify the coordinates of each vertex. Use the translation rule to find the coordinates of each vertex of the image.

( x , y ) → ( x − 2 , y − 5 ) P ( 2 , 1 ) → ( 2 − 2 , 1 − 5 ) , or  P ′ ( 0 , − 4 ) . Q ( 3 , 3 ) → ( 3 − 2 , 3 − 5 ) , or  Q ′ ( 1 , − 2 ) . R ( − 1 , 3 ) → ( − 1 − 2 , 3 − 5 ) , or  R ′ ( − 3 , − 2 ) . table with 4 rows and 1 column , row1 column 1 , open x comma y close rightwards arrow open x minus 2 comma y minus 5 close , row2 column 1 , p open 2 comma 1 close rightwards arrow open 2 minus 2 comma 1 minus 5 close comma or , p prime , open 0 comma negative 4 close . , row3 column 1 , q open 3 comma 3 close rightwards arrow open 3 minus 2 comma 3 minus 5 close comma or , q prime , open 1 comma negative 2 close . , row4 column 1 , r open negative 1 comma 3 close rightwards arrow open negative 1 minus 2 comma 3 minus 5 close comma or , r prime , open negative 3 comma negative 2 close . , end table

To graph the image of Δ P Q R , cap delta p q r comma  first graph P ′ , Q ′ , and  R ′ . p prime , comma , q prime , comma , and , r prime , .  Then draw P ′ Q ′ ¯ , Q ′ R ′ ¯ , p prime , q prime bar . comma . q prime , r prime bar . comma  and R ′ P ′ ¯ . r prime , p prime bar . .

✓ Got It?

3. 1. What are the images of the vertices of Δ A B C cap delta eh b c  for the translation ( x , y ) → ( x + 1 , y − 4 ) ? open x comma y close rightwards arrow . open x plus 1 comma y minus 4 close question mark  Graph Δ A B C cap delta eh b c  and its image.
   2. Reasoning Draw A A ′ ¯ , B B ′ ¯ , eh , eh prime bar . comma . b , b prime bar . comma  and C C ′ ¯ . c , c prime bar . .  What relationships exist among these three segments? How do you know?