Download presentation

1
Geometry 7.2 Reflections

2
**Goals Identify and use reflections in a plane.**

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

3
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

4
**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

5
**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

6
**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

7
**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

8
**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

9
**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

10
**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

11
**Reflect AB on the line x = 2.**

Tuesday, Dec 1, 1:58 PM 7.2 Reflections

12
Applications Tuesday, Dec 1, 1:58 PM 7.2 Reflections

13
**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

14
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

15
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

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

28
**Heron’s Solution Reflect one of the points over the line.**

Tuesday, Dec 1, 1:58 PM 7.2 Reflections

29
**Heron’s Solution Connect the other point to the reflected one.**

Tuesday, Dec 1, 1:58 PM 7.2 Reflections

30
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

31
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

32
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

33
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

34
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

35
**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

36
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

37
Line of Symmetry Tuesday, Dec 1, 1:58 PM 7.2 Reflections

38
Lines of Symmetry Tuesday, Dec 1, 1:58 PM 7.2 Reflections

39
**How many lines of symmetry?**

Two None Tuesday, Dec 1, 1:58 PM 7.2 Reflections

40
**Classical Architecture**

Tuesday, Dec 1, 1:58 PM 7.2 Reflections

41
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

42
Homework Facial Symmetry Tuesday, Dec 1, 1:58 PM 7.2 Reflections

Similar presentations

OK

Math 310 Sections 12.1-2 Isometry. Transformations Def A transformation is a map from the plane to itself that takes each point in the plane to exactly.

Math 310 Sections 12.1-2 Isometry. Transformations Def A transformation is a map from the plane to itself that takes each point in the plane to exactly.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google