Post on 03-Jan-2016
Lesson 12.5Applications of
Isometriespp. 522-525
Lesson 12.5Applications of
Isometriespp. 522-525
Objective:To apply isometries to the solutions of specific problems of a practical nature.
Objective:To apply isometries to the solutions of specific problems of a practical nature.
Principles of reflection apply in miniature golf, billiards, and bowling. Lenses for everything from cameras
to telescopes also involve reflections. Light always bounces off something at the same angle that it arrived. The
angle of incidence is equal to the angle of reflection.
Principles of reflection apply in miniature golf, billiards, and bowling. Lenses for everything from cameras
to telescopes also involve reflections. Light always bounces off something at the same angle that it arrived. The
angle of incidence is equal to the angle of reflection.
EXAMPLE 1 A beam will be sent from satellite A to satellite B by being sent to the earth and reflected by a booster station to satellite B. The satellite engineers are trying to place the booster station in the spot where the total distance that the beam travels will be the shortest. If the booster station must be located somewhere along line h, what is the best location?
EXAMPLE 1 A beam will be sent from satellite A to satellite B by being sent to the earth and reflected by a booster station to satellite B. The satellite engineers are trying to place the booster station in the spot where the total distance that the beam travels will be the shortest. If the booster station must be located somewhere along line h, what is the best location?
BB
B′B′
SSTThh
AA
EXAMPLE 2 Figure 12.17 shows a miniature golf green. Notice that it would be impossible to putt a ball directly from the tee (T) to the hole (H). What spots should you aim for on sides 1 and 2 so that you will make a hole in one?
EXAMPLE 2 Figure 12.17 shows a miniature golf green. Notice that it would be impossible to putt a ball directly from the tee (T) to the hole (H). What spots should you aim for on sides 1 and 2 so that you will make a hole in one?
T
H
side 1side 1
side
2sid
e 2
Practice ProblemsPractice Problems
1. Draw the shot on the miniature golf hole that would produce a hole in one by hitting exactly 2 parallel sides.
1. Draw the shot on the miniature golf hole that would produce a hole in one by hitting exactly 2 parallel sides.
2 1
2. Draw the shot on the miniature golf hole that would produce a hole in one by hitting 3 sides.
2. Draw the shot on the miniature golf hole that would produce a hole in one by hitting 3 sides.
3 2
3. Draw the shot on the miniature golf hole that would produce a hole in one by hitting 5 sides.
3. Draw the shot on the miniature golf hole that would produce a hole in one by hitting 5 sides.
4. Two towns on opposite sides of a river are to be joined by a road that takes the shortest path; the new bridge across the river must be perpendicular to the shorelines. Write a plan for finding the location of the bridge and draw a diagram showing its location.
4. Two towns on opposite sides of a river are to be joined by a road that takes the shortest path; the new bridge across the river must be perpendicular to the shorelines. Write a plan for finding the location of the bridge and draw a diagram showing its location.
AA
BB
PP
QQB′B′
Homeworkpp. 524-525Homeworkpp. 524-525
►A. Exercises1. Find the point on k that marks the
shortest distance from A to C.
►A. Exercises1. Find the point on k that marks the
shortest distance from A to C.
EE
AA BB
CC
DD
kk
►A. Exercises2. Explain why E is not the answer to
exercise 1.
►A. Exercises2. Explain why E is not the answer to
exercise 1.
The shortest distance between two points is a lineThe shortest distance between two points is a line
EE
AA BB
CC
DD
kk
►A. Exercises3. Find the point G on k that marks
the shortest distance from A to kto D.
►A. Exercises3. Find the point G on k that marks
the shortest distance from A to kto D.
EE
AA BB
CC
DDGG kk
►A. Exercises4. Compare AG + GD to AE + ED.
►A. Exercises4. Compare AG + GD to AE + ED.
EE
AA BB
CC
DDGG kk
EE
AA BB
CC
DDGG
►A. Exercises4. Compare AG + GD to AE + ED.
►A. Exercises4. Compare AG + GD to AE + ED.
kk
►A. Exercises5. Find the point on k that marks
the shortest distance from D to kto B.
►A. Exercises5. Find the point on k that marks
the shortest distance from D to kto B.
EE
AA BB
CC
DD
kk
►A. ExercisesAn electron moves from point X to Y by bouncing off sides of a four-sided enclosure. Find the path it will follow if it bounces off the given side(s).
6. DC
►A. ExercisesAn electron moves from point X to Y by bouncing off sides of a four-sided enclosure. Find the path it will follow if it bounces off the given side(s).
6. DC
XX
AA BB
CCDD
►A. Exercises
6. DC
►A. Exercises
6. DC
YY
►A. ExercisesAn electron moves from point X to Y by bouncing off sides of a four-sided enclosure. Find the path it will follow if it bounces off the given side(s).
7. AD
►A. ExercisesAn electron moves from point X to Y by bouncing off sides of a four-sided enclosure. Find the path it will follow if it bounces off the given side(s).
7. AD
XX
AA BB
CCDD
YY
►A. Exercises
7. AD
►A. Exercises
7. AD
XX
AA BB
CCDD
YY
►A. Exercises
8. AB and then BC
►A. Exercises
8. AB and then BC
XX
AA BB
CCDD
YY
►A. Exercises
9. AD and then DC
►A. Exercises
9. AD and then DC
XX
AA BB
CCDD
YY
►A. Exercises
10. AB and then DC
►A. Exercises
10. AB and then DC
■ Cumulative Review21. What is the fixed point of a
rotation called?
■ Cumulative Review21. What is the fixed point of a
rotation called?
■ Cumulative Review22. How many fixed points does a
translation have?
■ Cumulative Review22. How many fixed points does a
translation have?
■ Cumulative Review23. How many fixed points does a
reflection have?
■ Cumulative Review23. How many fixed points does a
reflection have?
■ Cumulative Review24. If T is a transformation that is a
dilation and X is the fixed point, what is T(X)?
■ Cumulative Review24. If T is a transformation that is a
dilation and X is the fixed point, what is T(X)?
■ Cumulative Review25. If an isometry has three
noncollinear fixed points, what can you conclude?
■ Cumulative Review25. If an isometry has three
noncollinear fixed points, what can you conclude?