Triangle ABC is given where B=61°, b=32, and a=35. How many distinct triangles can be made with the given measurements? Explain your answer.
To determine the number of distinct triangles that can be made with the given measurements, we can use the Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C)
Given:
B = 61°
b = 32
a = 35
We can first find angle A using the Law of Sines:
35/sin(A) = 32/sin(61°)
sin(A) = 35*sin(61°)/32
sin(A) ≈ 0.866
This gives us angle A ≈ 60°.
Now, we can find angle C using the fact that the sum of angles in a triangle is 180°:
C = 180° - A - B
C = 180° - 61° - 60°
C = 59°
So, we have two possible triangles that can be formed with the given measurements:
1. Triangle ABC with angles A ≈ 60°, B = 61°, and C = 59°
2. Triangle ACB with angles A ≈ 60°, B = 61°, and C = 59°
Therefore, a total of 2 distinct triangles can be made with the given measurements.