# Geometry 7.2 Reflections.

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Geometry 7.2 Reflections

Goals Identify and use reflections in a plane.
Understand Line Symmetry Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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 is its own reflection.) P P’ P and P’ are equidistant from line m. Line of Reflection Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Reflections on the Coordinate Plane
Graph the reflection of A(2, 3) in the x-axis. A(2, 3)  A’(2, -3) A Reflection in the x-axis has the mapping: (x, y)  (x, -y) 3 3 A’(2, -3) Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Reflections on the Coordinate Plane
Graph the reflection of A(2, 3) in the y-axis. 2 2 A(2, 3)  A’(-2, 3) A Reflection in the y-axis has the mapping: (x, y)  (-x, y) A’(-2, 3) Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Reflections on the Coordinate Plane
Graph the reflection of A(1, 4) in the line y = x. A(1, 4)  A’(4, 1) A Reflection in the line y = x has the mapping: (x, y)  (y, x) A’(4, 1) Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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. Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Reflect RST in y-axis. Determine coordinates. Mapping Formula:
(x, y)  (-x, y) R(0, 4)  R’(0, 4) S(-4, 1)  S’(4, 1) T(-1, -2)  T’(1, -2) R’(0, 4) R (0, 4) S (-4, 1) S’(4, 1) T T’(1, -2) (-1, -2) Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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) D(2, 2) B(-3, -1) C(3, -1) Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Other Reflections Any line can be used as the line of reflection.
Mapping formulas can be found, but for now counting is easier. Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Reflect AB on the line x = 2.
Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Applications Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Heron’s Problem Heron of Alexandria (10 – 70 AD)
Inventor of first steam engine. Wrote Dioptra, a collection of constructions to measure lengths from a distance. Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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). Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Heron’s Solution Reflect one of the points over the line.
Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Heron’s Solution Connect the other point to the reflected one.
Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Heron’s Solution The intersection of this line and the road is where the sum of the segments is a minimum. Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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. Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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 Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Heron’s Problem 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… B(4, 3) A(-4, 1) A’(-4, -1) Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Heron’s Problem 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). B(4, 3) A(-4, 1) B’(4, -3) Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Heron’s Problem AC + CB is a minimum. B(4, 3) A(-4, 1) C(-2, 0)
Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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. Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Line of Symmetry Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Lines of Symmetry Tuesday, Dec 1, 1:58 PM 7.2 Reflections

How many lines of symmetry?
Two None Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Classical Architecture
Tuesday, Dec 1, 1:58 PM 7.2 Reflections

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. Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Homework Facial Symmetry Tuesday, Dec 1, 1:58 PM 7.2 Reflections