5. In a right-angled triangle, \(a\) and \(b\) are the legs; \(S\) is the area of the triangle. Find the angles of the triangle if it is known that \((a+b)^{2}=8 S\)
5. Solution. Noting that \(8 S=8 \cdot 0.5 a b=4 a b\), and using the condition of the problem, we can write: \((a+b)^{2}-4 a b=0\), \((a-b)^{2}=0\), and therefore \(a=b\). This means that the right triangle is isosceles, and its angles are respectively \(45^{\circ}, 45^{\circ}\) and \(90^{\circ}\).