Lesson 12.5 Applications of Isometries pp Lesson 12.5 Applications of Isometries pp

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.

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?

B B B′ S S T T h h A A

EXAMPLE 2 Figure 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 1 side 2

Practice Problems

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. 3 2

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.

A A B B P P Q Q B′B′ B′B′

Homework pp Homework pp

►A. Exercises 1.Find the point on k that marks the shortest distance from A to C. ►A. Exercises 1.Find the point on k that marks the shortest distance from A to C. E E A A B B C C D D k k

►A. Exercises 2.Explain why E is not the answer to exercise 1. ►A. Exercises 2.Explain why E is not the answer to exercise 1. The shortest distance between two points is a line E E A A B B C C D D k k

►A. Exercises 3.Find the point G on k that marks the shortest distance from A to k to D. ►A. Exercises 3.Find the point G on k that marks the shortest distance from A to k to D. E E A A B B C C D D G G k k

►A. Exercises 4.Compare AG + GD to AE + ED. ►A. Exercises 4.Compare AG + GD to AE + ED. E E A A B B C C D D G G k k

E E A A B B C C D D G G ►A. Exercises 4.Compare AG + GD to AE + ED. ►A. Exercises 4.Compare AG + GD to AE + ED. k k

►A. Exercises 5.Find the point on k that marks the shortest distance from D to k to B. ►A. Exercises 5.Find the point on k that marks the shortest distance from D to k to B. E E A A B B C C D D k k

►A. Exercises An 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. Exercises An 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

X X A A B B C C D D ►A. Exercises 6.DC ►A. Exercises 6.DC Y Y

►A. Exercises An 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. Exercises An 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

X X A A B B C C D D Y Y ►A. Exercises 7.AD ►A. Exercises 7.AD

X X A A B B C C D D Y Y ►A. Exercises 8.AB and then BC ►A. Exercises 8.AB and then BC

X X A A B B C C D D Y Y ►A. Exercises 9.AD and then DC ►A. Exercises 9.AD and then DC

X X A A B B C C D D Y Y ►A. Exercises 10.AB and then DC ►A. Exercises 10.AB and then DC

■ Cumulative Review 21.What is the fixed point of a rotation called? ■ Cumulative Review 21.What is the fixed point of a rotation called?

■ Cumulative Review 22.How many fixed points does a translation have? ■ Cumulative Review 22.How many fixed points does a translation have?

■ Cumulative Review 23.How many fixed points does a reflection have? ■ Cumulative Review 23.How many fixed points does a reflection have?

■ Cumulative Review 24.If T is a transformation that is a dilation and X is the fixed point, what is T(X)? ■ Cumulative Review 24.If T is a transformation that is a dilation and X is the fixed point, what is T(X)?

■ Cumulative Review 25.If an isometry has three noncollinear fixed points, what can you conclude? ■ Cumulative Review 25.If an isometry has three noncollinear fixed points, what can you conclude?