The perimeter of triangle ABC is 20 + 4√5, the area is 16√5, and the measures of angles A, B, and C are approximately 36°, 76°, and 68°, respectively.
Perimeter:
AB = 12 (given)
AC = 8 (given)
BC = √(12^2 - 8^2) = √(144 - 64) = √80 = 4√5 (using the Pythagorean theorem)
Perimeter = AB + AC + BC = 12 + 8 + 4√5 = 20 + 4√5
Area:
Area of a triangle = 1/2 * base * height
In this case, we can use AC as the base (8) and BC as the height (4√5)
Area = 1/2 * 8 * 4√5 = 16√5
Angle measures:
We can use the Law of Sines to find the measures of all angles.
The Law of Sines states that the ratio of the side lengths of a triangle is equal to the ratio of the sines of the opposite angles.
In triangle ABC:
sin(A) = BC/AC = 4√5 / 8
sin(B) = AC/BC = 8 / 4√5 = 2√5 / 5
sin(C) = AB/BC = 12 / 4√5 = 3√5 / 5
Using the calculator and rounding to the nearest degree:
A ≈ 36°
B ≈ 76°
C ≈ 68°
Therefore, the perimeter of triangle ABC is 20 + 4√5, the area is 16√5, and the measures of angles A, B, and C are approximately 36°, 76°, and 68°, respectively.