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Geometry 7.2 Reflections

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Tuesday, Dec 1, 1:58 PM7.2 Reflections2 Goals Identify and use reflections in a plane. Understand Line Symmetry

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Tuesday, Dec 1, 1:58 PM7.2 Reflections3 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: 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 is its own reflection.) Line of Reflection m P P P and P are equidistant from line m.

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Tuesday, Dec 1, 1:58 PM7.2 Reflections4 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)

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Tuesday, Dec 1, 1:58 PM7.2 Reflections5 Reflections on the Coordinate Plane Graph the reflection of A(2, 3) in the y-axis. 22 A(-2, 3) A(2, 3) A(-2, 3) A Reflection in the y-axis has the mapping: (x, y) (-x, y)

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Tuesday, Dec 1, 1:58 PM7.2 Reflections6 Reflections on the Coordinate Plane Graph the reflection of A(1, 4) in the line y = x. A(4, 1) A(1, 4) A(4, 1) A Reflection in the line y = x has the mapping: (x, y) (y, x)

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Tuesday, Dec 1, 1:58 PM7.2 Reflections7 Reflection Mappings In the x-axis: (x, y) (x, -y) In the y-axis: (x, y) (-x, y) In y = x: (x, y) (y, x) 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.

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Tuesday, Dec 1, 1:58 PM7.2 Reflections8 Reflect RST in y-axis. R S T Determine coordinates. Mapping Formula: (x, y) (-x, y) R(0, 4) S(-4, 1) S(4, 1) T(-1, -2) T(1, -2) (0, 4) (-4, 1) (-1, -2) T(1, -2) S(4, 1) R(0, 4)

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Tuesday, Dec 1, 1:58 PM7.2 Reflections9 Reflect ABCD in the x-axis. Mapping Formula: (x, y) (x, -y) A(-2, 2) A(-2, -2) B(-3, -1) B(-3, 1) C(3, -1) C(3, 1) D(2, 2) D(2, -2) A(-2, 2) B(-3, -1)C(3, -1) D(2, 2)

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Tuesday, Dec 1, 1:58 PM7.2 Reflections10 Other Reflections Any line can be used as the line of reflection. Mapping formulas can be found, but for now counting is easier.

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Tuesday, Dec 1, 1:58 PM7.2 Reflections11 Reflect AB on the line x = 2. A(1, 3) B(0, 1) A(3, 3) B(4, 1)

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Tuesday, Dec 1, 1:58 PM7.2 Reflections12 Applications

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Tuesday, Dec 1, 1:58 PM7.2 Reflections13 Herons Problem Heron of Alexandria (10 – 70 AD) Inventor of first steam engine. Wrote Dioptra, a collection of constructions to measure lengths from a distance.

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Tuesday, Dec 1, 1:58 PM7.2 Reflections14 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections15 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections16 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections17 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections18 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections19 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections20 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections21 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections22 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections23 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections24 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections25 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections26 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections27 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections28 Herons Solution Reflect one of the points over the line.

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Tuesday, Dec 1, 1:58 PM7.2 Reflections29 Herons Solution Connect the other point to the reflected one.

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Tuesday, Dec 1, 1:58 PM7.2 Reflections30 Herons Solution The intersection of this line and the road is where the sum of the segments is a minimum.

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Tuesday, Dec 1, 1:58 PM7.2 Reflections31 Herons Solution The intersection of this line and the road is where the sum of the segments is a minimum. Put the box there.

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Tuesday, Dec 1, 1:58 PM7.2 Reflections32 Herons 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 b c

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Tuesday, Dec 1, 1:58 PM7.2 Reflections33 Herons Problem A(-4, 1) B(4, 3) A(-4, -1) 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…

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Tuesday, Dec 1, 1:58 PM7.2 Reflections34 Herons 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).

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Tuesday, Dec 1, 1:58 PM7.2 Reflections35 Herons Problem A(-4, 1) B(4, 3) C(-2, 0) AC + CB is a minimum.

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Tuesday, Dec 1, 1:58 PM7.2 Reflections36 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.

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Tuesday, Dec 1, 1:58 PM7.2 Reflections37 Line of Symmetry

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Tuesday, Dec 1, 1:58 PM7.2 Reflections38 Lines of Symmetry

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Tuesday, Dec 1, 1:58 PM7.2 Reflections39 How many lines of symmetry? Two None

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Tuesday, Dec 1, 1:58 PM7.2 Reflections40 Classical Architecture

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Tuesday, Dec 1, 1:58 PM7.2 Reflections41 Summary A point and its 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.

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Tuesday, Dec 1, 1:58 PM7.2 Reflections42 Homework Facial Symmetry

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