Final answer:
The lengths of the sides of the right triangle are x = 8/3, 2x - 1 = 13/3, and 2x + 1 = 19/3.
Step-by-step explanation:
The lengths of the sides of the right triangle are as follows:
- Leg 1: x
- Leg 2: 2x - 1
- Hypotenuse: 2x + 1
To find the lengths of the legs and hypotenuse, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
Using the Pythagorean theorem, we have:
x^2 + (2x - 1)^2 = (2x + 1)^2
Expanding, simplifying, and solving for x, we get:
x^2 + 4x^2 - 4x + 1 = 4x^2 + 4x + 1
3x^2 - 8x = 0
x(3x - 8) = 0
x = 0 or x = 8/3
Since the length of a side cannot be zero, we take x = 8/3.
Therefore, the lengths of the sides are:
- Leg 1: x = 8/3
- Leg 2: 2x - 1 = 2(8/3) - 1 = 16/3 - 1 = 13/3
- Hypotenuse: 2x + 1 = 2(8/3) + 1 = 16/3 + 1 = 19/3