Tuesday, Dec 1, 1:58 PM 7.2 Reflections 2
Goals
Identify and use reflections in a plane.
Understand Line Symmetry
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ReflectionA 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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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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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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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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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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Reflect RST in y-axis.
R
S
T
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)
(0, 4)
(-4, 1)
(-1, -2)T’(1, -2)
S’(4, 1)
R’(0, 4)
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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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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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Reflect AB on the line x = 2.
A(1, 3)
B(0, 1)
A’(3, 3)
B’(4, 1)
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Applications
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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.
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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Heron’s Solution
Reflect one of the points over the line.
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Heron’s Solution
Connect the other point to the reflected one.
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Heron’s SolutionThe intersection of this line and the road is where the sum of the segments is a minimum.
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Heron’s SolutionThe 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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Heron’s ExplanationThe 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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Heron’s 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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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).
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Heron’s Problem
A(-4, 1)
B(4, 3)
C(-2, 0)
AC + CB is a minimum.
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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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Line of Symmetry
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Lines of Symmetry
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How many lines of symmetry?
Two
None
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Classical Architecture
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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.
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Homework
Facial Symmetry