The sides of the triangle are 7; 15; 13. Find the corner opposite the middle side.

To calculate the angle opposite to the middle side, you can use the cosine theorem, according to which the square of one of the sides of the triangle is equal to the sum of the squares of the other two sides minus the double product of these sides by the cosine of the angle between them:

a ^ 2 = b ^ 2 + c ^ 2 – 2bccos α.

Let’s say AB = 7 cm; BC = 15 cm; AC = 13. The angle opposite to the middle side, in this case, is the angle ∠B.

AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 AB BC cos B;

2 AB BC cos B = AB ^ 2 + BC ^ 2 – AC ^ 2;

cos B = (AB ^ 2 + BC ^ 2 – AC ^ 2) / (2 AB BC);

cos B = (7 ^ 2 + 15 ^ 2 – 13 ^ 2) / 2 · 7 · 15 = (49 + 225 – 169) / 210 = 105/210 = ½, which corresponds to 60 °.

Answer: The angle opposite to the middle side is 60 °.