The Ambiguous SSA Triangle - Given an Acute Angle
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The Ambiguous SSA Triangle – Given an Acute Angle
Suppose you are given \(\triangle ABC\), where \(\angle A = 35^\circ\), \(b = 10\), and the length of side \(a\) is also given. Because you have an angle and its opposite length, the triangle can be solved using the sine law. However, four cases can occur given this information. These cases depend entirely on the given length of \(a\).
Case 1: \(a = b \sin A\)
The triangle is right angled when \(a = b \sin A = 10\sin 35^\circ\). This triangle can be solved by using primary trigonometric ratios. There will be only one solution.
Case 2: \(a < b \sin A\)
There is no triangle possible if \(a < b \sin A = 10 \sin 35^\circ\). For example, if \(a\) is \(3\) units long, then Figure 2 matches the case description. Notice that if \(a\) pivots about angle \(C\) and swings in an arc, that arc does not intersect side \(c\). Therefore, a triangle is not made. There is no solution.
Case 3: \(a > b\)
The length of \(a > b\). Any example of this case will have one distinct solution. In Figure 3, the given length of \(a\) is \(11\) units. Because \(11 > 10\), it fits the case description. If the arm \(a\) pivots at angle \(C\), you can see that it will intersect side \(c\) only at angle \(B\), making \(\triangle ABC\). This triangle can be solved using the sine law.
Case 4: \(b \sin A < a < b\)
In this situation, \(b \sin A < a < b\). Any example in this case has two distinct solutions. Since \(10 \sin 35 \doteq 5.74\) and \(b = 10\), \(a\) could be between approximately \(5.74\) units and \(10\) units. In Figure 4, the given length of \(a\) is \(8\) units. Because \(5.74 < 8 < 10\), it fits the case description. Again, if arm \(a\) pivots about angle \(C\), it will intersect side \(c\) at \(B'\) and \(B''\). That means there are two possible triangles, \(\triangle AB'C\) and \(\triangle AB''C\), and thus two solutions.
Key Lesson Marker |
Given a SSA triangle such as \(\triangle ABC\), and angle \(A\) is acute:
| Situation | Number of solutions
|
|
\(a \ge b\)
|
One solution |
|
\(a = h = b \sin A\)
|
One solution (right triangle)
|
|
\[\begin{align}
a &< h \\ a &< b \sin A\\ \end{align}\] |
No solution
|
|
\(b \sin A < a < b\)
|
Two solutions |