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

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

Let us introduce the notation. The side to be found is denoted with. Known sides a = 5 cm, b = 7 cm. The angle opposite to the side c is denoted by γ. If you know the two sides of the triangle and the angle between them, then the third side can be found by applying the cosine theorem. The square of any side of a triangle is equal to the sum of the squares of the other two sides, without twice the product of these sides by the cosine of the angle between them.

c ^ 2 = a ^ 2 + b ^ 2 – 2ab * cosγ;

c ^ 2 = 5 ^ 2 + 7 ^ 2 – 2 * 5 * 7 * cos 45;

c ^ 2 = 25 + 49 – 70 * √2 / 2;

c ^ 2 = 74 – 35√2 ≈ 74 – 35 * 1.4 = 74 – 49 = 25.

c ≈ 5.

Answer. five.