The Ambiguous Case: Two Triangles from the Same Facts

0:00Can the same facts about a triangle give two different triangles?

0:05Angle A is 35 degrees, side b is 10 centimetres, and side a, opposite angle A, is 7 centimetres.

0:14Side a must reach the base line, so swing it like a gate hinged at C.

0:20It meets the line twice, at B one and B two, so two triangles fit the same facts.

0:27Two sides and an angle not between them is the ambiguous case: more than one triangle can fit.

0:35To find angle B, use the sine law: sine B over b equals sine A over a.

0:42So sine B equals 10 times sine 35 degrees over 7, about 0.819.

0:50The inverse sine key turns the sine back into an angle: B is about 55 degrees.

0:57But 180 minus 55 is 125 degrees, and its sine is also 0.819.

1:06Both fit with A: 35 plus 55 is 90, and 35 plus 125 is 160, both under 180 degrees.

1:18At B one the angle is 55 degrees; at B two it is 125 degrees, an obtuse angle, over 90.

1:28A common mistake is keeping only the calculator's angle and missing a triangle.

1:34With A acute, under 90 degrees, compare a with the height from C, which is b times sine A, about 5.74 centimetres.

1:45If a is shorter than the height, there is no triangle; if equal, one right triangle.

1:52If a is between the height and b, like our 7 centimetres, there are two; if a is at least b, just one.

2:01In the ambiguous case, always check both B and 180 degrees minus B.