Answer:
a ≈ 10, b ≈ 11.6, c ≈ 18.3
Explanation:
To solve the triangle, we can use the Law of Sines. This law relates the ratios of the lengths of the sides of a triangle to the sines of their opposite angles.
The Law of Sines states:
a/sin(A) = b/sin(B) = c/sin(C)
Given the information provided, we know that angle A is 39°, angle B is 61°, and side a is 10.
To solve for the remaining side and angles, we can use the Law of Sines. Let's start by finding the ratio between side b and sin(B):
b/sin(B) = a/sin(A)
Substituting the given values:
b/sin(61°) = 10/sin(39°)
Now, we can solve for b by cross-multiplying:
b = (10 * sin(61°)) / sin(39°)
Using a calculator, we find that b ≈ 11.6 (rounded to the nearest tenth).
Next, let's find the remaining angle, angle C:
c/sin(C) = a/sin(A)
Substituting the given values:
c/sin(C) = 10/sin(39°)
Solving for sin(C):
sin(C) = (10 * sin(39°)) / c
Since the sum of the angles in a triangle is 180°, we can find angle C by subtracting angles A and B from 180°:
C = 180° - A - B
Substituting the given values:
C = 180° - 39° - 61°
C = 80°
Now, we can find side c using the Law of Sines:
c/sin(C) = a/sin(A)
Substituting the given values:
c/sin(80°) = 10/sin(39°)
Solving for c:
c = (10 * sin(80°)) / sin(39°)
Using a calculator, we find that c ≈ 18.3 (rounded to the nearest tenth).
Therefore, the triangle can be solved and the approximate lengths of the sides are:
a ≈ 10, b ≈ 11.6, c ≈ 18.3
Remember to always check the given information to ensure that the triangle can be solved using the Law of Sines.