Find the area and perimeter of a right-angled triangle if the smaller of its legs is 18 cm

Find the area and perimeter of a right-angled triangle if the smaller of its legs is 18 cm less than the hypotenuse, and the difference in legs is 17 cm.

Suppose that the length of the hypotenuse of this triangle is c, one leg is a and the smaller leg is b.

Then, by the condition of the problem, we get:

b – a = 17,

b = a + 17.

a = c – 18,

c = a + 18.

Let’s use the Pythagorean theorem for this triangle:

a² + b² = c²,

a² + (a + 17) ² = (a + 18) ²,

a² + a² + 34 * x + 289 = a² + 36 * x + 324,

a² – 2 * x – 35 = 0,

D = (-2) ² – 4 * 1 * (-35) = 144.

x = (2 + 12) / 2 = 7.

Since the solution to the problem can only be a positive number, this is the only root.

This means that the hypotenuse of the triangle is 7 + 18 = 25, and the second leg is 7 + 17 = 24.

Thus, the perimeter of the triangle is:

P = 7 + 24 + 25 = 56 (cm).

The area is equal to:

S = 7 * 24/2 = 84 (cm²).