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How many triangles exist that fit the following criteria?B = 30°, a = 4, b = 3

How many triangles exist that fit the following criteria?B = 30°, a = 4, b = 3-example-1

1 Answer

7 votes

Given:

B = 30°, a = 4, b = 3

We will solve the triangle to find how many triangles exist that fit the given data

We will use the sine rule to find the angle A as follows:


\begin{gathered} (a)/(sin(A))=(b)/(sin(B)) \\ \\ (4)/(sin(A))=(3)/(sin(30)) \\ \\ sin(A)=(4)/(3)sin(30)=(2)/(3) \\ \end{gathered}

So, the measure of angle A will be as follows:


A=sin^(-1)((2)/(3))=41.81\degree,or,138.19\degree

Now, we will find the measure of angle C using the fact that the sum of the angles = 180


\begin{gathered} C=180-(A+B) \\ A=41.81\degree\rightarrow C=180-(30+41.81)=108.19\operatorname{\degree} \\ A=138.19\operatorname{\degree}\rightarrow C=180-(30+138.19)=11.81\operatorname{\degree} \end{gathered}

So, there are two triangles that can fit the given data

So, the answer will be Two

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