The two sides of the triangle are 5 cm and 7 cm, and the angle opposite the third side is 45 °. Find the third side of the triangle?

To find the third side of a given triangle, we use the cosine theorem, according to which the equality holds for any triangle:
a ^ 2 + b ^ 2 – 2 * a * b * cos (α) = c ^ 2,
where a, b and c are the lengths of the sides of this triangle, and α is the angle of the triangle opposite to the side with.
According to the condition of the problem, in this triangle a = 5 cm, d = 7 cm, α = 45 °, therefore, we can write:
c ^ 2 = 5 ^ 2 + 7 ^ 2 – 2 * 5 * 7 * cos (45 °) = 25 + 49 – 70 * cos (45 °) = 74 – 70 * √2 / 2 = 74 – 35√2 …
Therefore, c = √ (74 – 35√2) see.
Answer: the third side of the triangle is √ (74 – 35√2) cm.