Indicate if there are 0, 1, or 2 triangles with these data.
1) A=30∘,b=10,a=12:
2) A=30∘,b=10,a=3
To determine if there are 0, 1, or 2 triangles with the given data, we can use the triangle inequality theorem.
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side.
For the first set of data:
1) A = 30°, b = 10, a = 12
We need to check if the sum of any two sides is greater than the third side.
b + a = 10 + 12 = 22, which is greater than the third side (b). So, one condition is satisfied.
a + b = 12 + 10 = 22, which is greater than the third side (b). So, the second condition is also satisfied.
Now, we need to check for the third side.
b + a = 10 + 12 = 22, which is greater than the third side (b).
Since all three conditions are satisfied, a triangle can be formed. Thus, there is 1 triangle.
For the second set of data:
2) A = 30°, b = 10, a = 3
We need to check if the sum of any two sides is greater than the third side.
b + a = 10 + 3 = 13, which is less than the third side (b). So, the first condition is not satisfied.
a + b = 3 + 10 = 13, which is less than the third side (b). So, the second condition is also not satisfied.
Since one or both of the conditions are not satisfied, a triangle cannot be formed. Thus, there are 0 triangles.
In summary:
1) A = 30°, b = 10, a = 12: There is 1 triangle.
2) A = 30°, b = 10, a = 3: There are 0 triangles.