One leg of a right triangle is 5 cm longer than the other leg. The hypotenuse is 25 cm long. Determine the measures of the legs of the triangle.
Let's call one leg of the right triangle x cm. Then the other leg is (x+5) cm.
We can use the Pythagorean theorem to solve for x:
x^2 + (x+5)^2 = 25^2
x^2 + x^2 + 10x + 25 = 625
2x^2 + 10x - 600 = 0
Divide both sides of the equation by 2:
x^2 + 5x - 300 = 0
Factor the quadratic equation:
(x + 20)(x - 15) = 0
Thus, the possible solutions are x = -20 or x = 15.
Since the length of a leg can't be negative, x = 15 cm.
So one leg of the triangle is 15 cm and the other leg is 15 cm + 5 cm = 20 cm. Answer: \boxed{15 \text{ cm}, 20 \text{ cm}}.
Let's denote the length of one leg as x cm. Since the other leg is 5 cm longer, its length can be represented as x + 5 cm.
According to the Pythagorean theorem, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
So, we have the equation:
x^2 + (x + 5)^2 = 25^2
Expanding the equation:
x^2 + (x^2 + 10x + 25) = 625
Combining like terms:
2x^2 + 10x + 25 = 625
Subtracting 625 from both sides:
2x^2 + 10x + 25 - 625 = 0
2x^2 + 10x - 600 = 0
Dividing both sides by 2 to simplify the equation:
x^2 + 5x - 300 = 0
Now we can solve this quadratic equation. Factoring or using the quadratic formula, we find the solutions:
x = 15 or x = -20
Since length cannot be negative in this context, we discard the negative solution.
Therefore, one leg of the triangle is 15 cm long. And since the other leg is 5 cm longer, it is 15 + 5 = 20 cm long.
So, the measures of the legs of the triangle are 15 cm and 20 cm.