In each of the three different cases, I constructed three triangles with a 120 degree angle but different relationships between the lengths of the sides.
In Case 1, where line BC was longer than line AB, I discovered that one triangle could be formed while keeping angle A at 120 degrees.
In Case 2, where line BC was equal to line AB, it was impossible for line segment AC to form since line segment BC was not long enough to connect with the dashed line. Therefore, no triangles were formed.
In Case 3, where line BC was shorter than line AB, the same problem in Case 2 arose. Line segment BC was not long enough to connect with the dashed line. Again, no triangles were formed.
In conclusion, I found out that in order for an obtuse triangle to be created, the following rules apply*:
• If BC > AB, then one triangle is possible.
• If BC = AB, then no triangles are possible.
• If BC < AB, then no triangles are possible.
*Note: The lettering scheme I use may be different than the lettering schemes used byothers. However, the same concepts I mention here still apply.
In my constructions:
• Line segment BC refers to the side opposite of the obtuse angle measuring 120 degrees
• Line segment AB refers to the leg of the angle measuring 120 degrees which touches line segment BC as mentioned above
These three cases here were based on acute triangles, and the relationships between the lengths of the sides and the height of the triangle. In each of the cases, in order for the triangle to still have an angle A measuring 30 degrees, point C of line segment BC would have had to connect with the line segment.
In Case 1, line segment BC was shorter than the triangle’s “minimum” height, so therefore no triangles were constructed. The dashed red circle I drew represents the fact that no matter where point C is placed, line BC will never connect with the dashed line.
In Case 2, line segment BC was equal to the height, so one triangle was constructed.
In Case 3, line segment BC was greater than line segment AB which was greater than the height. One triangle was possible.
In conclusion, in an acute triangle:
• If BC < h, then no triangles are possible.
• If BC = h, then one triangle is possible.
• If BC > AB > h, then one triangle is possible.
In Exploration 3, I drew a triangle where line AB was greater than line BC which was greater than the triangle’s height. I discovered that two triangles were possible while keeping an angle A measuring 60 degrees. The triangle in Exploration 3 is similar to the triangles in Exploration 2. If this triangle were an Exploration 2 triangle, then point C would be able to connect to the dashed line in two places, therefore creating two triangles that we can see here. Basically, there is enough room for line BC to exist in two places.
In conclusion, if in an acute triangle:
• If AB > BC > h, then two triangles are possible.
Questions
1. Explain why sin C is the same in both triangles in the ambiguous case (exploration 3). This is why the Law of Sines is also ambiguous in this case.
Let’s recall the values for the triangle in Exploration 3. Line AB equals 8 cm (in the Law of Sines formula, this would be side c), angle A equals 60 degrees, line BC equals 7 ½ cm (LoS formula: side a), and the height equals 7 cm.
The Law of Sines is ambiguous in this case since two different triangles will share part of the same Law of Sines formula since they have a congruent angle and two sides (otherwise known as the misleading SSA congruency conjecture). In the triangle in Exploration 3:
sin 60 = sin C
7.5 8
The Law of Sines here misleads us into thinking there is only one triangle, when two are possible. If one were to solve this equation they would only get one of the triangles. It would require one to solve the sin B/b part (which is where the ambiguity comes in), to get the full picture.
2. Explain from diagram 4 in Exploration 2 why a unique triangle is determined if BC < AB.
In acute triangles, a triangle can be constructed when BC < AB since point C of line BC has enough length to “connect” to AB. However, two triangles are not possible since unlike in Exploration 3, there is not enough “room” for point C to connect in two places without point C overshooting line AB.
3. What did you discover about the number of triangles you can make given SSA (two sides and an angle)? Summarize the "rules" for all cases.
Given SSA, the case in Exploration 3, two triangles can be made.
All in all:
- If in an obtuse triangle:
- If BC > AB, then one triangle is possible.
- If BC = AB, then no triangles are possible.
- If BC < AB, then no triangles are possible.
- If in an acute triangle:
- If BC < h, then no triangles are possible.
- If BC = h, then one triangle is possible.
- If BC > AB > h, then one triangle is possible.
- If AB > BC > h, then two triangles are possible.
Since other people use different lettering schemes, I will list how my line segments correspond to the Law of Sines formula:
• AB = c
• BC = a
• AC = b
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