Find the legs of a right-angled triangle if their sum is 46 cm and the hypotenuse is 34 cm.

Let’s denote the legs of this right-angled triangle by a and b. It is known that their sum is 46 cm, therefore, the following ratio is true:
a + b = 46.
Also, according to the condition of the problem, the hypotenuse of this right-angled triangle is 34 cm, therefore, using the Pythagorean theorem, we can write:
a ^ 2 + b ^ 2 = 34 ^ 2.
We solve the resulting system of equations. Substituting into the first equation the value b = 46 – and from the second equation, we get:
a ^ 2 + (46 – a) ^ 2 = 1156.
We solve the resulting equation:
a ^ 2 + a ^ 2 – 92 * a + 2116 = 1156;
2 * a ^ 2 – 92 * a + 960 = 0;
a ^ 2 – 46 * a + 480 = 0.
The discriminant of this quadratic equation is 46 ^ 2 – 4 * 480 = 14 ^ 2. Therefore, the roots of this equation are:
a1 = 0.5 * (46 – 14) = 16;
a2 = 0.5 * (46 + 14) = 30.
Knowing a, we find the second leg b:
b1 = 46 – a1 = 46 – 16 = 30;
b2 = 46 – a2 = 46 – 30 = 16.

Answer: the legs of this right-angled triangle are 16 cm and 30 cm.