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

Let’s denote the length of the smaller of the legs of this right-angled triangle by x. Then the length of the second leg is x + 1.

According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the legs:

25 = x ^ 2 + (x + 1) ^ 2.

Solving a quadratic equation
Let’s open the brackets and solve the quadratic equation:

x ^ 2 + x ^ 2 + 2x + 1 = 25;

2x ^ 2 + 2x – 24 = 0;

x ^ 2 + x – 12 = 0;

By the theorem, conversely to Vieta’s theorem, x1 = – 4, x2 = 3.

Since the length cannot be a negative number, only the second root of the equation satisfies the condition of the problem.

Finding the area of ​​a triangle
So, we have established that the length of the smaller leg is 3. Then the length of the second leg is 4. Find the area of ​​this right-angled triangle:

S = 0.5 * (3 + 4) = 3.5.

Answer: 3.5.