Find the missing parts of the triangle.
C = 124.2°
a = 6.90 km
b = 11.60 km
use the law of cosines to start and find side c. Then use the law of sines to find remaining angles.
To find the missing parts of the triangle, we can use the Law of Sines since we are given an angle and the lengths of two sides.
The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. In other words, we have the following proportion:
sin(A) / a = sin(B) / b = sin(C) / c
Given that C = 124.2°, a = 6.90 km, and b = 11.60 km, we can solve for the missing part, which is side c.
First, let's find the measure of angle A:
A = 180° - B - C
A = 180° - 90° - 124.2°
A = 180° - 214.2°
A ≈ -34.2°
Since angles in triangles cannot be negative, we need to take the positive supplement of -34.2°:
A = 180° - (-34.2°)
A = 180° + 34.2°
A ≈ 214.2°
Now, we can use the Law of Sines to find side c:
sin(A) / a = sin(C) / c
Plugging in the values we know:
sin(214.2°) / 6.90 km = sin(124.2°) / c
Now, let's solve for c:
c = (sin(124.2°) * 6.90 km) / sin(214.2°)
Calculating this expression, we can find the value of c.
c ≈ 6.14 km
Therefore, the missing part of the triangle, side c, is approximately 6.14 km.