Geometry

7.2 Reflections

Tuesday, Nov 18, 2:51 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).

Tuesday, Nov 18, 2:51 PM 7.2 Reflections 16

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, Nov 18, 2:51 PM 7.2 Reflections 17

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, Nov 18, 2:51 PM 7.2 Reflections 18

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, Nov 18, 2:51 PM 7.2 Reflections 19

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, Nov 18, 2:51 PM 7.2 Reflections 20

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, Nov 18, 2:51 PM 7.2 Reflections 21

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, Nov 18, 2:51 PM 7.2 Reflections 22

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, Nov 18, 2:51 PM 7.2 Reflections 23

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, Nov 18, 2:51 PM 7.2 Reflections 24

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, Nov 18, 2:51 PM 7.2 Reflections 25

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, Nov 18, 2:51 PM 7.2 Reflections 26

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, Nov 18, 2:51 PM 7.2 Reflections 27

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.