What is the maximum number of distinct triangles that can be formed if m<A = 30, b = 8, and a = 5
(1) 1 (3) 3
(2) 2 (4) 0
Can I use law of sines to answer such questions about how many different triangles?
Hello Magentarita,
Let's look at both cases: Recall that the sine of an angle is equal to the sine of its supplement.
Case I (diagram 1)
$\displaystyle \frac{\sin A}{A}=\frac{\sin B}{B}$
$\displaystyle \frac{\sin 30}{5}=\frac{\sin B}{8}$
$\displaystyle \sin B = .8$
$\displaystyle B \approx 53.1^{\circ}$
Case II (diagram 2)
$\displaystyle \triangle B'BC$ is isosceles.
$\displaystyle \angle B$ is supplementary to $\displaystyle \angle AB'C$
$\displaystyle \angle AB'C=180-53.1 \Rightarrow 126.9$
$\displaystyle \frac{\sin 30}{5}=\frac{\sin 126.9}{8}=\frac{\sin 53.1}{8}$
So, looks like 2 triangles can have the characteristics you describe.
$\displaystyle \triangle ABC \ \ and \ \ \triangle AB'C$