Determine whether the triangle has no/one/two solutions. Then Find and round measures of sides to the nearest tenth and measures of the angles to the nearest degree.
A= 50 degrees, a=34, b=40
Given: A = 50 Deg., a = 34, b = 40.
sinB/b = sinA/a.
sinB/40 = sin50/34.
sinB = 40(sin50/34) = 0.9012.
B = 64 Deg.
C = 180 - A - B = 180 - 50 - 64=66 Deg.
c/sinC = a/sinA.
c/sin64 = 40/sin50.
c = sin64(40/sin50) = 46.9.
To determine whether the triangle has no/one/two solutions, we can use the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we have the measures of the angles and sides as follows:
A = 50 degrees
a = 34
b = 40
Let's name the missing side as c.
Now, let's check if the Triangle Inequality Theorem is satisfied for all the sides:
1. For sides a and b: a + b > c
Substituting the given values: 34 + 40 > c
74 > c
2. For sides a and c: a + c > b
Substituting the given values: 34 + c > 40
c > 6
3. For sides b and c: b + c > a
Substituting the given values: 40 + c > 34
c > -6
From the above calculations, we see that side c must be greater than 6 units. However, there are no other restrictions on its length since side c can be as long as needed.
Therefore, the triangle has ONE solution since there is a range of possible lengths for side c.
Next, let's find the measures of the sides and angles:
1. Measure of side c:
Since side c can have a range of lengths, we cannot find a specific value. However, we can express it as c > 6 units.
2. Measure of angle B:
To find angle B, we can use the Law of Sines, which states that the ratio of a side length to the sine of its corresponding angle is the same for all pairs of sides and angles in a triangle.
Applying the Law of Sines: sin(B) / b = sin(A) / a
Substituting the given values: sin(B) / 40 = sin(50 degrees) / 34
Solving for sin(B): sin(B) = (40 / 34) * sin(50 degrees)
sin(B) ≈ 0.797
Taking the inverse sine, we find: B ≈ 51.5 degrees (rounded to the nearest degree)
3. Measure of angle C:
Since angle C is the remaining angle, we can find its measure by subtracting the sum of angles A and B from 180 degrees: C = 180 - A - B
Substituting the given values: C = 180 - 50 - 51.5
C ≈ 78.5 degrees (rounded to the nearest degree)
4. Rounding measures of sides to the nearest tenth:
Side a: 34 (already rounded to a whole number)
Side b: 40 (already rounded to a whole number)
Side c: Since c > 6 units, we can round it to the nearest tenth as c ≈ 6.0 units.
In summary, the triangle has one solution. The measures of the sides, rounded to the nearest tenth, are a = 34, b = 40, and c ≈ 6.0 units. The measures of the angles, rounded to the nearest degree, are A = 50 degrees, B ≈ 51.5 degrees, and C ≈ 78.5 degrees.