4To solve a right triangle, you can use the Pythagorean Theorem and/or basic trig ratios.CABabc
5In an oblique triangle (one in which there are no right angles) neither ofthose methods will work.Solving an oblique triangle requiresusing either the Law of Sines orthe Law of Cosines, depending onwhich information you are given.
6Law of SinesThe Law of Sines can be usedwhen you are given two anglesand one side of an oblique triangle(ASA or AAS).
7The Law of Sines can also be used to solve an oblique trianglewhen you are given two sidesand the angle opposite one ofthe sides (SSA).
8Suppose you are given the measures shown below for two sides and an angle ofan oblique triangle.abA
9The pieces could be put together to form this triangle. But they could alsobe put together toform this triangle.bAabAa
10Since SSA does not always define a unique triangle, this bAabAaSince SSA does not alwaysdefine a unique triangle, thisis called the Ambiguous Caseof the Law of Sines.
11We are going to create a map problem involving the ambiguous case wheretwo unique triangles are possible withthe same given measures.Students could work on this projectindividually or in pairs.
12On the map you are given, find two cities with identical names (1 and 2). Hometown 1Hometown 2
13Use a ruler to draw a line through the two cities. Hometown 1Hometown 2
14Locate another city on the same line – preferably not between the two cities (A).Hometown 1Hometown 2Abbaville
15Construct the perpendicular bisector of the segment between cities 1 and 2.Hometown 1Hometown 2Abbaville
16perpendicular bisector (B). Locate a city on theperpendicular bisector (B).Hometown 1Hometown 2BigtownAbbaville
17Use a ruler to draw a line connecting cities A and B. Find the distance between thosecities by measuring and usingthe map’s scale.Hometown 1Hometown 2BigtownAbbaville
18Verify by measuring that cities 1 and 2 are approximately the same distancefrom city B.Why is that true?Hometown 1Hometown 2BigtownAbbaville
19Verify by measuring that cities 1 and 2 are approximately the same distancefrom city B.Why is that true?Hometown 1Hometown 2BigtownAbbavilleRemember a theorem from geometry which says that any point on the perpendicular bisector of a segment is equidistant from its endpoints.
20Use the map’s scale to compute the distance from city B to each of cities 1 and 2.Hometown 1Hometown 2BigtownAbbaville
21Use a piece of patty paper (or other method) to draw perpendicular axes to represent directions(N-S-E-W). Place the origin ofthe axes on city A.Hometown 1Hometown 2BigtownAbbaville
22Use a protractor to find angle bearings to city B and to either city 1 or city 2 (the same bearings).Hometown 1Hometown 2BigtownAbbaville
23Using 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.Hometown 1Hometown 2BigtownAbbaville
24How far is Abbaville from Hometown? 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?
25Remember that there are two solutions to your 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.
26I would have each student (or pair of students) turn in all work they had doneon the activity.I would also have them turn in one sheetof paper that contains only the statementof their problem. This would be used toexchange problems with others in theclass, giving students more practice insolving problems with the Law of Sinesand to check accuracy and clear statementof their problems.