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Ambiguous Map Problem (application of the Ambiguous Case of the Law of Sines)

How Far to Where?

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How Far to Where?. Ambiguous Map Problem (application of the Ambiguous Case of the Law of Sines). Students should have done some work with the Ambiguous Case of the Law of Sines before doing this project. Review of Ambiguous Case of the Law of Sines. To solve a right triangle, you can - PowerPoint PPT Presentation

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Page 1: How Far to Where?

Ambiguous Map Problem

(application of the Ambiguous Case of the Law of Sines)

Page 2: How Far to Where?

Students should have done somework with the Ambiguous Case ofthe Law of Sines before doingthis project.

Page 3: How Far to Where?

Review of Ambiguous Case of theLaw of Sines

Page 4: How Far to Where?

To solve a right triangle, you canuse the Pythagorean Theoremand/or basic trig ratios.

C

A

Ba

bc

2 2 2a b c

asinA

cb

cos Aca

tanAb

ccsc A

ac

sec Abb

cot Aa

Page 5: How Far to Where?

In an oblique triangle (one in which there are no right angles) neither of those methods will work.

Solving an oblique triangle requires using either the Law of Sines orthe Law of Cosines, depending onwhich information you are given.

Page 6: How Far to Where?

Law of Sines a b c

sinA sinB sinC

The Law of Sines can be used when you are given two anglesand one side of an oblique triangle(ASA or AAS).

Page 7: How Far to Where?

The Law of Sines can also be used to solve an oblique trianglewhen you are given two sidesand the angle opposite one ofthe sides (SSA).

Page 8: How Far to Where?

b a

A

Suppose you are given the measures shown below fortwo sides and an angle ofan oblique triangle.

Page 9: How Far to Where?

b

A

a

The pieces could be put together to form thistriangle.

b

A

a

But they could alsobe put together to form this triangle.

Page 10: How Far to Where?

Since SSA does not alwaysdefine a unique triangle, thisis called the Ambiguous Caseof the Law of Sines.

b

A

ab

A

a

Page 11: How Far to Where?

We are going to create a map probleminvolving the ambiguous case wheretwo unique triangles are possible withthe same given measures.

Students could work on this project individually or in pairs.

Page 12: How Far to Where?

Hometown 1

Hometown 2

On the map you are given, find two cities with identical names (1 and 2).

Page 13: How Far to Where?

Hometown 1

Hometown 2

Use a ruler to draw a line through the two cities.

Page 14: How Far to Where?

Hometown 1

Hometown 2

Abbaville

Locate another city on the sameline – preferably not between thetwo cities (A).

Page 15: How Far to Where?

Hometown 1

Hometown 2

Construct the perpendicular bisectorof the segment between cities 1 and 2.

Abbaville

Page 16: How Far to Where?

Hometown 1

Hometown 2

Locate a city on the perpendicular bisector (B).

AbbavilleBigtown

Page 17: How Far to Where?

Hometown 1

Hometown 2

Use a ruler to draw a line connecting cities A and B. Find the distance between those cities by measuring and using the map’s scale.

AbbavilleBigtown

Page 18: How Far to Where?

Hometown 1

Hometown 2

Verify by measuring that cities 1 and 2 are approximately the same distancefrom city B.

Why is that true?

AbbavilleBigtown

Page 19: How Far to Where?

Hometown 1

Hometown 2

Verify by measuring that cities 1 and 2 are approximately the same distancefrom city B.

Why is that true?

AbbavilleBigtown

Remember a theorem from geometry which says that any point on the perpendicular bisector of a segment is equidistant from its endpoints.

Page 20: How Far to Where?

Hometown 1

Hometown 2

Use the map’s scale to compute the distancefrom city B to each of cities 1 and 2.

AbbavilleBigtown

Page 21: How Far to Where?

Hometown 1

Hometown 2

Use a piece of patty paper (or other method) todraw perpendicular axes to represent directions(N-S-E-W). Place the origin of the axes on city A.

AbbavilleBigtown

Page 22: How Far to Where?

Hometown 1

Hometown 2

Use a protractor to find angle bearings to city Band to either city 1 or city 2 (the same bearings).

AbbavilleBigtown

Page 23: How Far to Where?

Hometown 1

Hometown 2

Using the measurements you have calculated,create a problem asking for the distance fromcity A to city 1 (or 2). This is an ambiguousquestion with two unique solutionsbecause two cities havethe same name.

AbbavilleBigtown

Page 24: How Far to Where?

State your problem – in a creative way if you can. Be sure you include the basic facts, something like the following.

Hometown is 50 degrees east of north of Abbaville.Bigtown is 86 degrees east of north of Abbaville.Bigtown is 48 miles from Abbaville.Hometown is 20 miles from Bigtown.

How far is Abbaville from Hometown?

Page 25: How Far to Where?

Remember that there are two solutions toyour question because there are two citieswith the same name (Hometown in myexample).

Work out both solutions, using the Law ofSines. Then check your solutions withaccurate ruler measures and your map’sscale.

Page 26: How Far to Where?

I would have each student (or pair ofstudents) turn in all work they had doneon the activity.

I would also have them turn in one sheetof paper that contains only the statement of their problem. This would be used to exchange problems with others in theclass, giving students more practice insolving problems with the Law of Sines and to check accuracy and clear statementof their problems.