Determine if there are zero, one, or two triangles for the following:
mZA=48°
a = 10 m
b = 12 m
To determine if there are zero, one, or two triangles, we can use the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Given:
mZA = 48°
a = 10 m
b = 12 m
Let's denote the third side as c:
c = a * sin(ZA) / sin(180 - ZA)
c = 10 * sin(48) / sin(132)
c ≈ 7.525 m
Now, we can check if there are zero, one, or two triangles:
1. a + b > c
10 + 12 > 7.525
22 > 7.525
This condition is satisfied, so there can be at least one triangle.
2. b + c > a
12 + 7.525 > 10
19.525 > 10
This condition is satisfied as well.
3. a + c > b
10 + 7.525 > 12
17.525 > 12
This condition is also satisfied.
Since all three conditions are satisfied, there can be one triangle that satisfies the given conditions.