Your team has just won the flag in a flag football tournament at your school. As a reward, you get to take home the flag and keep it until the next game, when the other team will try to win it back. The flag looks like this:
It makes an isosceles triangle. You start to wonder how many different possible triangles there are for different lengths of sides. For example, if you make an oblique triangle that has a given angle greater than ninety degrees, how many ways are there to do this? Can you determine how many different possible triangles there are if the triangle is an isosceles triangle?
SSA Triangles
In Geometry, you learned that two sides and a non-included angle do not necessarily define a unique triangle.
Consider the following cases given
Case 1: No triangle exists
In this case
Case 2: One triangle exists
In this case,
Case 3: Two triangles exist
In this case,
Case 4: One triangle exists
In this case
Case 5: One triangle exists
In this case,
Case 3 is referred to as the Ambiguous Case because there are two possible triangles and two possible solutions. One way to check to see how many possible solutions (if any) a triangle will have is to compare sides
If: | Then: | |
---|---|---|
a. | No solution, one solution, two solutions | |
i. | No solution | |
ii. | One solution | |
iii. | Two solutions | |
b. | One solution | |
c. | One solution |
Identifying Triangles
For the following problems, determine if the sides and angles given determine no, one or two triangles.
1. The set contains an angle, its opposite side and the side between them.
2. The set contains an angle, its opposite side and the side between them.
Even though
3. The set contains an angle, its opposite side and the side between them.
Even though
Examples
Example 1
Earlier, you were given a problem about a triangle.
As you now know, when two sides of a triangle with an included side are known, and the lengths of the two sides are equal, there is one possible solution. Since an isosceles triangle meets these criteria, there is only one possible solution.
Example 2
Determine how many solutions there would be for a triangle based on the given information and by calculating
Example 3
Determine how many solutions there would be for a triangle based on the given information and by calculating
Example 4
Determine how many solutions there would be for a triangle based on the given information and by calculating \begin{align*}b \sin A\end{align*} and comparing it with \begin{align*}a\end{align*}. Sketch an approximate diagram for each problem in the box labeled “diagram.”
\begin{align*}A = 47.8^\circ, a = 13.48,b = 18.2\end{align*}
\begin{align*}A = 47.8^\circ, a = 13.48,b = 18.2\end{align*} \begin{align*}13.48 = 13.48\end{align*} 1 solution
Review
Determine if the sides and angle given determine no, one or two triangles. The set contains an angle, its opposite side and another side of the triangle.
- \begin{align*}a = 6, b = 6, A = 45^\circ\end{align*}
- \begin{align*}a = 4, b = 7, A = 115^\circ\end{align*}
- \begin{align*}a = 5, b = 2, A = 68^\circ\end{align*}
- \begin{align*}a = 7, b = 6, A = 34^\circ\end{align*}
- \begin{align*}a = 5, b = 3, A = 89^\circ\end{align*}
- \begin{align*}a = 4, b = 4, A = 123^\circ\end{align*}
- \begin{align*}a = 6, b = 8, A = 57^\circ\end{align*}
- \begin{align*}a = 4, b = 9, A = 24^\circ\end{align*}
- \begin{align*}a = 12, b = 11, A = 42^\circ\end{align*}
- \begin{align*}a = 15, b = 17, A = 96^\circ\end{align*}
- \begin{align*}a = 9, b = 10, A = 22^\circ\end{align*}
- In \begin{align*}\triangle ABC\end{align*}, a=4, b=5, and \begin{align*}m\angle A=32^\circ\end{align*}. Find the possible value(s) of c.
- In \begin{align*}\triangle DEF\end{align*}, d=7, e=5, and \begin{align*}m\angle D=67^\circ\end{align*}. Find the possible value(s) of f.
- In \begin{align*}\triangle KQD\end{align*}, \begin{align*}m\angle K=20^\circ\end{align*}, k=24, and d=31. Find \begin{align*}m\angle D\end{align*}.
- In \begin{align*}\triangle MRS\end{align*}, \begin{align*}m\angle M=70^\circ\end{align*}, m=44, and r=25. Find \begin{align*}m\angle R\end{align*}.
Review (Answers)
To see the Review answers, open this PDF file and look for section 5.9.