The perimeter of a right-angled triangle is 40 cm and one of the legs is 8 cm, find the second leg and its hypotenuse.

Let us take the leg of a right-angled triangle as x cm, and the hypotenuse as y cm. The perimeter of a triangle is equal to the sum of the lengths of its three sides, and is equal to (x + y + 8) cm or 40 cm. We obtain the equation x + y + 8 = 40. For a rectangular triangle we apply the Pythagorean theorem: The square of the hypotenuse is equal to the sum of the squares of the legs, i.e. y ^ 2 = x ^ 2 + 8 ^ 2. Let’s combine the equations into a system and solve it.

x + y + 8 = 40; y ^ 2 = x ^ 2 + 8 ^ 2;

x + y = 40 – 8; y ^ 2 = x ^ 2 + 64;

x + y = 32; y ^ 2 = x ^ 2 + 64 – we express from the first equation of the system the variable x through y;

x = 32 – y – substitute the expression (32 – y) in the second equation of the system instead of x;

y ^ 2 = (32 – y) ^ 2 + 64;

y ^ 2 = 1024 – 64y + y ^ 2 + 64;

y ^ 2 – y ^ 2 + 64y = 1024 + 64;

64y = 1088;

y = 1088: 64;

y = 17 (cm) – hypotenuse;

x = 32 – 17 = 15 (cm) – leg.

Answer. The hypotenuse is 17 cm, the leg is 15 cm.