B Apply

Use the given points and lines. Graph A B ¯ eh b bar  and its image A ′ B ′ ¯ eh prime , b prime bar  after a reflection first across ℓ 1 bold italic script l sub 1  and then across ℓ 2 . bold italic script l sub 2 , .  Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation.

24. A ( 1 , 5 )  and  B ( 2 , 1 ) ; ℓ 1 : x = 3 ; ℓ 2 : x = 7 eh open 1 comma 5 close , and b open 2 comma 1 close semicolon , script l sub 1 , colon x equals 3 semicolon , script l sub 2 , colon x equals 7
25. A ( 2 , 4 )  and  B ( 3 , 1 ) ; ℓ 1 : x - axis ; ℓ 2 : y - axis eh open 2 comma 4 close , and b open 3 comma 1 close semicolon , script l sub 1 , colon x minus , axis , semicolon , script l sub 2 , colon y minus , axis
26. A ( − 4 , − 3 )  and  B ( − 4 , 0 ) ; ℓ 1 : y = x ; ℓ 2 : y = − x eh open negative 4 comma negative 3 close , and b open negative 4 comma 0 close semicolon , script l sub 1 , colon y equals x semicolon , script l sub 2 , colon y equals negative x
27. A ( 2 , − 5 )  and  B ( − 1 , − 3 ) ; ℓ 1 : y = 0 ; ℓ 2 : y = 2 eh open 2 comma negative 5 close , and b open negative 1 comma negative 3 close semicolon , script l sub 1 , colon y equals 0 semicolon , script l sub 2 , colon y equals 2
28. A ( 6 , − 4 )  and  B ( 5 , 0 ) ; ℓ 1 : x = 6 ; ℓ 2 : x = 4 eh open 6 comma negative 4 close , and b open 5 comma 0 close semicolon , script l sub 1 , colon x equals 6 semicolon , script l sub 2 , colon x equals 4
29. A ( − 1 , 0 )  and  B ( 0 , − 2 ) ; ℓ 1 : y = − 1 ; ℓ 2 : y = 1 eh open negative 1 comma 0 close , and b open 0 comma negative 2 close semicolon , script l sub 1 , colon y equals negative 1 semicolon , script l sub 2 , colon y equals 1
30. Think About a Plan T → T ′ ( 1 , 5 ) t rightwards arrow , t prime , open 1 comma 5 close  by a glide reflection where the translation is ( x , y ) → ( x + 3 , y ) open x comma y close rightwards arrow open x plus 3 comma y close  and the line of reflection is y = 1 . y equals 1 .  What are the coordinates of T?
    • How can you work backwards to find the coordinates of T?
    • Should T be to left or to the right of T ′ ? t prime , question mark
    • Should T be above or below T ′ ? t prime , question mark

The two figures are congruent. Describe the isometry that maps the black figure onto the blue figure.

31. 
    Image Long Description
32. 

33. Which transformation maps the black triangle onto the blue triangle?

    

    1. a glide reflection where the translation is ( x , y ) → ( x , y − 3 ) open x comma y close rightwards arrow open x comma y minus 3 close  and the line of reflection is x = − 2 x equals , negative 2
    2. a 180 ° 180 degrees  rotation about the origin
    3. a reflection across the line y = − 1 2 y equals negative , 1 half
    4. a reflection across the y-axis followed by a 180 ° 180 degrees rotation about the origin
34. Writing Reflections and glide reflections are odd isometries, while translations and rotations are even isometries. Use what you have learned in this lesson to explain why these categories make sense.
35. Open-Ended Draw Δ A B C . cap delta eh b c .  Describe a reflection, a translation, a rotation, and a glide reflection. Then draw the image of Δ A B C . cap delta eh b c .  for each transformation.
36. Reasoning The definition states that a glide reflection is the composition of a translation and a reflection. Explain why these can occur in either order.

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