1. 4.2 Reflections Goals: Perform Reflections Perform Glide Reflections
Identify lines of symmetry
Solve real-life problems involving reflections

2. Core Concept Reflection
A reflection is a transformation that uses a line like a mirror to reflect a figure. The mirror line is called the line of reflection.
A reflection in a line m maps every point P in the plane to the point P’, so that for each point, one of the following properties is true:
-If P is not on m, then m is the perpendicular bisector of PP’, or
-If P is on m, then P =P’

3. Example 1a: Reflecting in Horizontal and Vertical Lines
Graph ∆𝐴𝐵𝐶 with vertices A(1,3), B(5,2) and C(2,1) and its image after the reflection… In the line n : x = 3 n A A’ B B’ C C’

4. Example 1b: Reflecting in Horizontal and Vertical Lines
Graph ∆𝐴𝐵𝐶 with vertices A(1,3), B(5,2) and C(2,1) and its image after the reflection… In the line m : y = 1 A B C’ C m B’ A’

5. You Try!
1) Graph ∆𝐴𝐵𝐶 from example 1 and its image after the reflection described. In the line j : x = 4 j A A’ B’ B C C’

6. Core Concept Coordinate Rules for Reflections
If (a,b) is reflected in the x-axis, then its image is the point (a,-b)
If (a,b) is reflected in the y-axis, then its image is the point (-a,b)
If (a,b) is reflected in the line y = x, then its image is the point (b,a)
If (a,b) is reflected in the line y = -x, then its image is the point (-b,-a)

7. Example 2a: Reflecting in the Line y = x
Graph FG with endpoints F(-1,2) and G(1,2) and its image after a reflection in the y = x.
If (a,b) is reflected in the line y = x, then its image is the point (b,a).
y = x
F F’(2,-1)
G G’(2, 1) F G G’ F’

8. Example 2b: Reflecting in the Line y = -x
Graph FG with endpoints F(-1,2) and G(1,2) and its image after a reflection in the y = x.
If (a,b) is reflected in the line y = -x, then its image is the point (-b,-a).
y = -x
F F’(-2,1)
G G’(-2, -1) F G F’ G’

9. You try!
The vertices of ∆𝐽𝐾𝐿 are J(1,3), K(4,4) and L(3,1). 2) Graph ∆𝐽𝐾𝐿 and its image after its reflection in the x-axis. 3) Graph ∆𝐽𝐾𝐿 and its image after its reflection in the line y = -x.
2) J’(1,-3), K’(4,-4), L’(3,-1)
3) J’(-3,-1), K’(-4,-4), L’(-1,-3) K J L L’ J’ L’ J’ K’ K’

10. Core Concept Reflection Postulate A reflections is a rigid motion
Glide Reflection
A transformation involving a translation followed by a reflection in which every point P is mapped to P”.

11. Example 3: Performing a Glide Reflection
Graph ∆𝐴𝐵𝐶 with vertices A(3,2), B(6,3) and C(7,1) and its image after the glide reflection. Translation: (x,y) (x-12,y) Reflection: in the x-axis B’ B A
A’(-9,2), B’(-6,3), C’(-5,1)
B”(-9,-2), B”(-6,-3), C”(-5,-1) A’ C C’ C” A” B”

12. Example 5: Identifying Lines of Symmetry
A figure in the plane has a line of symmetry when the figure can be mapped onto itself by a reflection in a line. This line of reflection is called a line of symmetry.
How many lines of symmetry does each hexagon have?

13. Example 6: Solving Real-Life Problems
Finding a Minimum Distance You are going to buy books. Your friend is going to buy CDs. Where should you park to minimize the distance you both should walk?
Reflect B in line m to obtain B’. Then draw AB’.
Label the intersection of AB’ and m as C.
Park at point C.