In a right-angled triangle, the hypotenuse is 5, and one of the legs is 1 larger than the other, find the area of the triangle.

Let’s find the lengths of the legs of this right-angled triangle.

Let’s denote by x the length of the larger leg of this right-angled triangle.

The problem statement says that one of the legs is 1 more than the other. Consequently, the smaller leg is 1 less than the larger leg and its length is x – 1.

According to the condition of the problem, the hypotenuse of this right-angled triangle is 5, therefore, using the Pythagorean theorem, we obtain the following equation:

x ^ 2 + (x – 1) ^ 2 = 5 ^ 2.

Solving this equation, we get:

x ^ 2 + x ^ 2 – 2x + 1 = 25;

2x ^ 2 – 2x + 1 – 25 = 0;

2x ^ 2 – 2x – 24 = 0;

x ^ 2 – x – 12 = 0;

x = (1 ± √ (1 + 4 * 9)) / 2 = (1 ± √49) / 2 = (1 ± 7) / 2;

x = (1 + 7) / 2 = 8/2 = 4.

We find the second leg:

x – 1 = 4 – 1 = 3.

Find the area of ​​the triangle:

4 * 3/2 = 12/2 = 6.

Answer: the area of ​​the triangle is 6.