Answer:
Given a triangle with one angle and two sides, it is possible to have multiple triangles with the same measurements. In this case, we have a triangle ABC with angle A equal to 42 degrees, side AB equal to 3, and side AC equal to 8. We need to determine how many distinct triangles can be formed with these measurements.
To find the third side of the triangle, we can use the law of cosines, which states that c^2 = a^2 + b^2 - 2abcos(C), where c is the side opposite to angle C, and a and b are the other two sides. Applying this formula, we get:
BC^2 = AB^2 + AC^2 - 2(AB)(AC)cos(A)
BC^2 = 3^2 + 8^2 - 2(3)(8)cos(42)
BC^2 ≈ 50.935
BC ≈ 7.139
Therefore, the length of side BC is approximately 7.139. Now we have all three sides of the triangle, namely AB = 3, AC = 8, and BC ≈ 7.139, and we can use these measurements to form different triangles.
However, we need to keep in mind that the angles of the triangle are not independent of its sides. In fact, the sum of the angles in a triangle is always 180 degrees, and therefore, once we fix two angles, the third angle is uniquely determined. In our case, we know that angle A is 42 degrees, but we do not know angles B and C. Therefore, we need to consider all possible values of angles B and C that satisfy the triangle inequality, which states that the sum of any two sides of a triangle must be greater than the third side.
Since we already know that AB = 3, AC = 8, and BC ≈ 7.139, we can use the law of cosines again to find the possible values of angles B and C for each triangle. We can write:
cos(B) = (AC^2 + BC^2 - AB^2) / (2ACBC)
cos(C) = (AB^2 + BC^2 - AC^2) / (2ABBC)
Plugging in the values, we get:
cos(B) ≈ 0.801
cos(C) ≈ -0.167
We can use the inverse cosine function to find the possible values of angles B and C:
B ≈ 36.63 degrees, 323.37 degrees
C ≈ 142.63 degrees, 217.37 degrees
However, we need to discard any values that do not satisfy the triangle inequality. Since angle A is already fixed at 42 degrees, we only need to check the sum of angles B and C for each triangle. We can write:
B + C = 180 - A
Plugging in the values, we get:
B + C ≈ 97.37 degrees
Therefore, the only possible values for angles B and C that satisfy the triangle inequality are B ≈ 36.63 degrees and C ≈ 97.37 degrees, or B ≈ 97.37 degrees and C ≈ 36.63 degrees. These two combinations of angles correspond to two distinct triangles, which are mirror images of each other.
In conclusion, we can form two distinct triangles with the given measurements of angle A = 42 degrees, side AB = 3, and side AC = 8. These triangles have angles A = 42 degrees, B ≈ 36.63 degrees, and C ≈ 97.37 degrees, or angles A = 42 degrees