Solve the triangle. Round answers to the nearest tenth. B = \boxed{\text{ } }^\circ, C = \boxed{\text{ } }^\circ, a = \boxed{\text{ } }
The Correct Answer and Explanation is:
Here is a step-by-step solution to solve the triangle.
1. Identify the given information:
From the image, we have a triangle with the following known values:
- Angle A = 96.7°
- Side b (the side opposite angle B) = 10.3 km
- Side c (the side opposite angle C) = 12.6 km
This is a Side-Angle-Side (SAS) case. We need to find the length of side ‘a’ and the measures of angles B and C.
2. Find the length of side ‘a’ using the Law of Cosines:
The Law of Cosines formula is: a² = b² + c² – 2bc cos(A)
- Substitute the known values into the formula:
a² = (10.3)² + (12.6)² – 2(10.3)(12.6)cos(96.7°) - Calculate the squares and the product:
a² = 106.09 + 158.76 – 259.56 * cos(96.7°) - Calculate the value of cos(96.7°):
cos(96.7°) ≈ -0.1167 - Continue the calculation for a²:
a² ≈ 264.85 – 259.56 * (-0.1167)
a² ≈ 264.85 + 30.28
a² ≈ 295.13 - Take the square root to find ‘a’:
a = √295.13 ≈ 17.179 - Round to the nearest tenth:
a = 17.2 km
3. Find the measure of angle B using the Law of Sines:
The Law of Sines formula is: a/sin(A) = b/sin(B)
We can rearrange this to solve for sin(B): sin(B) = (b * sin(A)) / a
- Substitute the known and calculated values (using the more precise value for ‘a’ for better accuracy):
sin(B) = (10.3 * sin(96.7°)) / 17.179 - Calculate the value of sin(96.7°):
sin(96.7°) ≈ 0.9932 - Continue the calculation for sin(B):
sin(B) ≈ (10.3 * 0.9932) / 17.179
sin(B) ≈ 10.22996 / 17.179
sin(B) ≈ 0.5955 - Find angle B by taking the inverse sine (arcsin):
B = arcsin(0.5955) ≈ 36.55° - Round to the nearest tenth:
B = 36.6°
4. Find the measure of angle C:
The sum of the angles in a triangle is 180°.
C = 180° – A – B
- Substitute the values for A and B:
C = 180° – 96.7° – 36.6°
C = 46.7° - C = 46.7°
Final Answer:
The solved values for the triangle, rounded to the nearest tenth, are:
- B = 36.6°
- C = 46.7°
- a = 17.2 km
