Answer: The measures of the angles in triangle ABC are BAC = 68°, BCA = 50.6°, and ABC = 61.4°. The lengths of the sides are AB = 12, AC = 14, and BC = 13.68.
Explanation:
To find the angles and sides of a triangle, you need to know at least three pieces of information about the triangle. In this case, you are given the lengths of sides AB and AC, and the measure of angle BAC.
You can use the Law of Sines to find the measure of the third angle and the length of the third side. The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of the angle opposite that side is the same for all sides and angles in the triangle. In other words, if we let a, b, and c represent the lengths of the sides of the triangle and A, B, and C represent the measures of the angles opposite those sides, then the following equation holds:
a/sin A = b/sin B = c/sin C
In triangle ABC, we are given the lengths of sides AB and AC and the measure of angle BAC. We can use the Law of Sines to find the measure of angle BCA and the length of side BC.
First, we can find the measure of angle BCA using the equation:
a/sin A = b/sin B = c/sin C
Substituting the known values, we get:
14/sin BAC = 12/sin BCA
Solving for sin BCA, we get:
sin BCA = 14/12 * sin BAC
Plugging in the given value for sin BAC, we get:
sin BCA = 14/12 * sin 68°
Using a calculator, we find that sin 68° = 0.906308
So sin BCA = 0.763696
To find the measure of angle BCA, we can use the inverse sine function on our calculator. The inverse sine of 0.763696 is about 50.6°.
Now that we know the measures of angles BAC and BCA, we can use the fact that the angles in a triangle sum to 180° to find the measure of angle ABC:
BAC + BCA + ABC = 180°
Substituting the known values, we get:
68° + 50.6° + ABC = 180°
Solving for ABC, we get:
ABC = 180° - 68° - 50.6°
This simplifies to:
ABC = 61.4°
Now that we have found the measures of all three angles, we can use the Law of Sines again to find the length of side BC.
We can use the equation:
a/sin A = b/sin B = c/sin C
Substituting the known values, we get:
14/sin BAC = 12/sin BCA = BC/sin ABC
Solving for BC, we get:
BC = 14/sin BAC * sin ABC
Plugging in the values we found earlier, we get:
BC = 14/0.906308 * sin 61.4°
Using a calculator, we find that sin 61.4° = 0.875
So BC = 14/0.906308 * 0.875 = 13.68
The measures of the angles in triangle ABC are BAC = 68°, BCA = 50.6°, and ABC = 61.4°. The lengths of the sides are AB = 12, AC = 14, and BC = 13.68.