# Math Help - Maximum Triangles

1. ## Maximum Triangles

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?

2. Originally Posted by magentarita
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)

$\frac{\sin A}{A}=\frac{\sin B}{B}$

$\frac{\sin 30}{5}=\frac{\sin B}{8}$

$\sin B = .8$

$B \approx 53.1^{\circ}$

Case II (diagram 2)

$\triangle B'BC$ is isosceles.

$\angle B$ is supplementary to $\angle AB'C$

$\angle AB'C=180-53.1 \Rightarrow 126.9$

$\frac{\sin 30}{5}=\frac{\sin 126.9}{8}=\frac{\sin 53.1}{8}$

So, looks like 2 triangles can have the characteristics you describe.

$\triangle ABC \ \ and \ \ \triangle AB'C$

3. ## Tell me......

Originally Posted by masters
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)

$\frac{\sin A}{A}=\frac{\sin B}{B}$

$\frac{\sin 30}{5}=\frac{\sin B}{8}$

$\sin B = .8$

$B \approx 53.1^{\circ}$

Case II (diagram 2)

$\triangle B'BC$ is isosceles.

$\angle B$ is supplementary to $\angle AB'C$

$\angle AB'C=180-53.1 \Rightarrow 126.9$

$\frac{\sin 30}{5}=\frac{\sin 126.9}{8}=\frac{\sin 53.1}{8}$

So, looks like 2 triangles can have the characteristics you describe.

$\triangle ABC \ \ and \ \ \triangle AB'C$
Is this the ambiguous case?

4. Originally Posted by magentarita
Is this the ambiguous case?
Yes, it is.

5. ## by the way...

Originally Posted by masters
Yes, it is.
By the way, I like playing with ambiguous case trig questions.