Geometry. 4.2 Reflections
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1 Geometry 4.2 Reflections
2 4.2 Warm Up 1. Write a rule for the translation of LMN to L M N. For #2-5, use the translation. (x, y) (x 8, y + 4) 2. What is the image of A(2, 6)? 3. What is the image of B( 1, 5)? 4. What is the preimage of C ( 3, 10)? 5. What is the preimage of D (4, 3)?
3 4.2 Essential Question How can you reflect a figure in a coordinate plane?
4 Goals Perform reflections in a plane. Perform glide reflecrtions. Identify lines of symmetry. Solve real-life problems involving reflections.
5 Reflection A reflection in line m is a transformation that maps every point P in the plane to point P so the following properties are true: m 1. If P is not on m, then m is the perpendicular bisector of PP. 2. If P is on m, then P = P. (The point P is its own reflection.) P P P and P are equidistant from line m. Line of Reflection
6 Postulate 4.2 Reflection Postulate A reflection is a rigid motion. Remember a rigid transformation is an isometry.
7 Reflections on the Coordinate Plane Graph the reflection of A(2, 3) in the x-axis. 3 3 A (2, -3) A(2, 3) A (2, -3) A Reflection in the x-axis has the mapping: x, y (x, y ) (a, b) (a, -b)
8 Reflections on the Coordinate Plane 2 A (-2, 3) 2 Graph the reflection of A(2, 3) in the y-axis. A(2, 3) A (-2, 3) A Reflection in the y-axis has the mapping: x, y (x, y ) (a, b) (-a, b)
9 Reflections on the Coordinate Plane Graph the reflection of A(1, 4) in the line y = x. A(1, 4) A (4, 1) A (4, 1) A Reflection in the line y = x has the mapping: x, y (x, y ) (a, b) (b, a)
10 Reflection Mappings x, y (x, y ) In the x-axis: a, b (a, b) In the y-axis: a, b ( a, b) In y = x: a, b (b, a) We say: Reflect in the x-axis, reflect over the x-axis, reflect on the x-axis, reflect across the x-axis. They mean the same thing.
11 Example 1 Reflect RST in y-axis. R R (0, 4) (0, 4) Determine coordinates. Mapping Formula: a, b ( a, b) S (-4, 1) T (-1, -2) T (1, -2) S (4, 1) R(0, 4) R (0, 4) S(-4, 1) S (4, 1) T(-1, -2) T (1, -2)
12 Example 2 Reflect ABCD in the x-axis. A(-2, 2) D(2, 2) B(-3, -1) C(3, -1) Mapping Formula: a, b ( a, b) A(-2, 2) A (-2, -2) B(-3, -1) B (-3, 1) C(3, -1) C (3, 1) D(2, 2) D (2, -2)
13 Example 3 Reflect FG in the Line y=x. F(-1, 2) G(1, 2) G (2, 1) Mapping Formula: a, b (b, a) F(-1, 2) F (2, -1) G(1, 2) G (2, 1) F (2, -1) Remember the line of reflection is the perpendicular bisector of FF, the segment connecting the preimage to the image.
14 Other Reflections Any line can be used as the line of reflection. Mapping formulas can be found, but for now counting is easier.
15 Example 4 Reflect AB on the line x = 2. A(1, 3) B(0, 1) A (3, 3) B (4, 1)
16 Glide Reflection A glide reflection is a transformation where a translation (the glide) is followed by a reflection. Line of Reflection
17 Glide Reflection 1. A translation maps P onto P. 2. A reflection in a line k parallel to the direction of the translation maps P to P. 1 2 Line of Reflection 3
18 Glide Reflection 1. A translation maps P onto P. 2. A reflection in a line k parallel to the direction of the translation maps P to P. 1 Line of Reflection
19 Compositions & Isometries If each transformation in a composition is an isometry, then the composition is an isometry. A Glide Reflection is an isometry.
20 Example 5 Find the image of ABC after a glide reflection. A(-4, 2), B(-2, 5), C(1, 3) Translation: (x, y) (x + 7, y) Reflection: in the x-axis
21 Find the image of ABC after a glide reflection. A(-4, 2), B(-2, 5), C(1, 3) Translation: (x, y) (x + 7, y) Reflection: in the x-axis (-2, 5) (-4, 2) (1, 3)
22 Find the image of ABC after a glide reflection. A(-4, 2), B(-2, 5), C(1, 3) Translation: (x, y) (x + 7, y) Reflection: in the x-axis (-2, 5) (5, 5) (-4, 2) (1, 3) (3, 2) (8, 3)
23 Find the image of ABC after a glide reflection. A(-4, 2), B(-2, 5), C(1, 3) Translation: (x, y) (x + 7, y) Reflection: in the x-axis (-2, 5) (5, 5) (-4, 2) (1, 3) (3, 2) (3, -2) (8, 3) (8, -3) (5, -5)
24 Find the image of ABC after a glide reflection. A(-4, 2), B(-2, 5), C(1, 3) Translation: (x, y) (x + 7, y) Reflection: in the x-axis (-2, 5) (5, 5) (-4, 2) Glide (1, 3) (3, 2) (8, 3) Reflection (3, -2) (8, -3) (5, -5)
25 You do it. Given the points M(-6, -6), N(-5, -2), O(-2, -1), and P(-3, -5), draw MNOP. Find the image of MNOP after a glide reflection. Translation: (x, y) (x, y + 7) Reflection: over y-axis. M N P O
26 You do it. Reflect over y-axis. N O O N M P P M N O M P
27 Example 6 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 A (-9, 2) B (-6, 3) A(3, 2) B(6, 3) C (-5, 1) C(7, 1) C (-5, -1) A (-9, -2) B (-6, -3)
28 Heron Heron of Alexandria (10 70 AD) Inventor of first steam engine. Wrote Dioptra, a collection of constructions to measure lengths from a distance.
29 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible). Place a point on the line where you think the box should be placed.
30 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible).
31 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible).
32 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible).
33 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible).
34 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible).
35 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible).
36 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible).
37 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible).
38 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible).
39 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible).
40 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible).
41 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible).
42 Heron s Problem The cable TV company wants to place a distribution box on the road so that the length of cable needed to go to both houses is a minimum (as small as possible).
43 Heron s Solution Reflect one of the points over the line.
44 Heron s Solution Connect the other point to the reflected one.
45 Heron s Solution The intersection of this line and the road is where the sum of the segments is a minimum.
46 Heron s Solution The intersection of this line and the road is where the sum of the segments is a minimum. Put the box there.
47 Heron s Explanation The sum of a + b is the shortest distance between the two points. b = c because the box is on the perpendicular bisector between the point and its reflection. So a + c is also a minimum. a c b
48 Now You try Heron s Problem A(-4, 1) A (-4, -1) B(4, 3) Find point C on the x-axis so that AC + CB is a minimum. 1. Reflect A in the x-axis. 2. Draw a line from A to B. 3. The line intersects the x- axis at C(-2, 0). Or
49 Now You try Heron s Problem A(-4, 1) B(4, 3) B (4, -3) Find point C on the x-axis so that AC + CB is a minimum. 1. Reflect B in the x-axis. 2. Draw a line from B to A. 3. The line intersects the x- axis at C(-2, 0).
50 Now You try Heron s Problem B(4, 3) AC + CB is a minimum. A(-4, 1) C(-2, 0)
51 Symmetry A similarity of form or arrangement on either side of a dividing line; correspondence of opposite parts in size, shape and position. Balance or beauty of form resulting from such correspondence. A figure that has line symmetry can be mapped onto itself.
52 Lines of Symmetry
53 Lines of Symmetry
54 Classical Architecture
55 Example 7 How many lines of symmetry does each hexagon have? Two Six One
56 Your Turn How many lines of symmetry? Two None Five
57 Summary A point and it s reflection are the same distance from the line of symmetry, but on opposite sides. Reflections are Isometries. A line of reflection is also a line of symmetry. A Glide-Reflection is a composition of a translation followed by a reflection.
58 Assignment Facial Symmetry
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