The perimeter of the triangle is 15 cm and the length of one side is 7 cm. Calculate the degree measure of the angle opposite

The perimeter of the triangle is 15 cm and the length of one side is 7 cm. Calculate the degree measure of the angle opposite this side if the bisector divides it in a ratio of 3: 5.

Let ABC be a triangle: BC = 7 cm, AB + BC + AC = 15 cm, VK / SK = 3/5.
1. The bisector of the inner corner of a triangle divides the side opposite to the corner into segments proportional to the other two sides:
VK / SK = AB / AC, then:
AB / AC = 3/5.
2. From the perimeter of triangle ABC:
AB + BC + AC = 15;
AB + 7 + AC = 15;
AB + AC = 15 – 7;
AB + AC = 8.
3. We have a system of linear equations with two unknowns (AB = x, AC = y):
x / y = 3/5;
x + y = 8.
In the second equation, we express x:
x = 8 – y.
Substitute in the first equation:
(8 – y) / y = 3/5;
5 (8 – y) = 3y (in proportion);
40 – 5y = 3y;
3y + 5y = 40;
8y = 40;
y = 40/8;
y = 5.
Thus, AC = y = 5 cm.
4. Find the length x:
x = 8 – y = 8 – 5 = 3.
Then, AB = x = 3 cm.
5. By the cosine theorem:
cosA = (AB ^ 2 + AC ^ 2 – BC ^ 2) / 2 * AB * AC;
cosA = (3 ^ 2 + 5 ^ 2 – 7 ^ 2) / 2 * 3 * 5 = (9 + 25 – 49) / 30 = – 15/30 = – 1/2.
So angle A is 120 degrees.
Answer: angle A = 120 degrees.