One of the legs of a right-angled triangle is 2 cm less than the second and the perimeter is 34 cm, find the area of the triangle.

1. Length of the first leg: A cm;

2. Length of the second leg: B cm;

3. Length of the hypotenuse: C cm;

4. By the condition of the problem:

A = B + 2;

A + B + C = 34 cm;

5. Hypotenuse:

C = 34 – (A + B) = 34 – (B + 2 + B) = 32 – 2 * B;

6. Pythagoras to help us:

A² + B² = C²;

(B + 2) ² + B² = (32 – 2 * B) ²;

2 * B² + 4 * B + 4 = 1024 – 128 * B + 4 * B²;

B² – 66 * B + 510 = 0;

B1,2 = 33 + – sqrt (33² -510) = 33 + – 24;

B1 = 33 + 24 = 57, (will not work, more than the perimeter);

B = 33 – 24 = 9 cm;

A = B + 2 = 9 + 2 = 11 cm;

7. Area of the triangle: S cm²;

S = (A * B) / 2 = (11 * 9) / 2 = 49.5 cm².

Answer: S = 49.5 cm².