Answer:
C. 2
Step-by-step explanation:
First, we have to understand that Uppercase variables denote angles, and lowercase variables denote side lengths.
Usually, corresponding upper and lowercase letters (e.g. A and a) are opposite from each other. Therefore, we can say that b = AC and a = BC.
See the attached image for a diagram.
If we draw this triangle, we can see that we are given angle-side-side. This is the ambiguous case of the Law of Sines. It tells us that two distinct triangles can be made with the given characteristics:
Using the Law of Sines and considering that side a is longer than side b and that angle B is acute, there is only one possible distinct triangle that can be formed, since an obtuse angle at A would violate the triangle inequality. Therefore, the correct answer is : (b).
To determine how many distinct triangles can be made with the given measurements, we must consider the Law of Sines which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
The key is to check whether the angle provided is acute (B = 61°) has a corresponding side that is longer or shorter than the other given side (a = 35 and b = 32). Given that the known angle B is acute, and side a is longer than side b, there is a possibility of two triangles if the angle opposite the longer side can be both acute and obtuse, but only one triangle if there is no such possibility.
The Law of Sines formula is ℓ = (a/sin A) = (b/sin B), where ℓ is the constant ratio, A and B are angles, and a and b are the lengths of the sides opposite these angles, respectively. However, since in this case a > b and angle B is not obtuse, we can have only one distinct triangle because an obtuse angle at A would violate the triangle inequality criterion.
Therefore, the answer is B. 1 distinct triangle.
