Final answer:
Using the Pythagorean theorem, we set up an equation with the given hypotenuse length of 15 cm and the relationship between the legs. Solving the resulting quadratic equation, we determined the lengths of the legs are 8 cm and 17 cm.
Step-by-step explanation:
To find the lengths of the legs of the right triangle, we will use the Pythagorean theorem, which states that a² + b² = c², where a and b are the legs of the triangle and c is the hypotenuse.
If one leg is 1 cm more than twice the length of the other leg, we can express this as a = 2b + 1.
The hypotenuse is given as 15 cm, so we can set up the equation as:
(2b + 1)² + b² = 15²
Solving for b, we get the following equation:
4b² + 4b + 1 + b² = 225
Combining like terms gives us:
5b² + 4b + 1 = 225
Now we subtract 225 from both sides to get a quadratic equation:
5b² + 4b - 224 = 0
Factoring the quadratic equation or using the quadratic formula, we find that b = 8.
Solving for a gives us a = 2(8) + 1 = 17.
Therefore, the lengths of the legs of the right triangle are 8 cm and 17 cm.