Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Each point is reflected across the line indicated. Find the coordinates of each image.

7. Q across x = 1 x equals 1
8. R across y = − 1 y equals , negative 1
9. S across the y-axis
10. T across y = 0.5 y equals 0.5
11. U across x = − 3 x equals , negative 3
12. V across the x-axis

See Problem 2.

Coordinate Geometry Given points J(1, 4), A(3, 5), and R(2, 1), graph Δ J A R cap delta j eh r  and its reflection image across each line.

13. the x-axis
14. the y-axis
15. y = 2 y equals 2
16. y = 5 y equals 5
17. x = − 1 x equals , negative 1
18. x = 2 x equals 2

See Problem 3.

19. Civil Engineering Town officials in Waterville and Drighton plan to construct a water pumping station along the Franklin Canal. The station will provide both towns with water. Where along the canal should the officials build the pumping station to minimize the total length of pipe needed?

    

B Apply

Copy each figure and line ℓ . bold italic script l .  Draw each figure's reflection image across line ℓ . bold italic script l .

20. 
21. 
    Image Long Description
22. 
23. Coordinate Geometry The following steps explain how to reflect point A across the line y = x . y equals x .
    • Step 1 Draw line ℓ script l  through A(5, 1) perpendicular to the line y = x . y equals x .  The slope of y = x y equals x  is 1, so the slope of line ℓ script l  is 1 · ( − 1 ) , 1 middle dot . open negative 1 close comma  or − 1 . negative 1 .
    • Step 2 From A, move two units left and two units up to y = x . y equals x .  Then move two more units left and two more units up to find the location of A ′ eh prime  on line ℓ . script l .  The coordinates of A ′ eh prime  are (1, 5).

    

    1. Copy the diagram. Then draw the lines through B and C that are perpendicular to the line y = x . y equals x .  What is the slope of each line?
    2. Reflect B and C across the line y = x . y equals x .  What are the coordinates of B ′ b prime  and C ′ ? c prime , question mark
    3. Graph Δ A ′ B ′ C ′ . cap delta , eh prime , b prime , c prime , .
    4. Make a Conjecture Compare the coordinates of the vertices of Δ A B C cap delta eh b c  and Δ A ′ B ′ C ′ . cap delta , eh prime , b prime , c prime , .  Make a conjecture about the coordinates of the point P(a, b) reflected across the line y = x . y equals x .
24. Coordinate Geometry Δ A B C cap delta eh b c  has vertices A ( − 3 , 5 ) , B ( − 2 , − 1 ) , and  C ( 0 , 3 ) . eh open negative 3 comma 5 close comma b open negative 2 comma negative 1 close comma , and c open 0 comma 3 close .  Graph Δ A B C cap delta eh b c  and Δ A ′ B ′ C ′ , cap delta , eh prime , b prime , c prime , comma  the reflection of Δ A B C cap delta eh b c  across the line y = − x . y equals . negative x .  (Hint: See Exercise 23.)

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