Final answer:
Using the Pythagorean theorem, the lengths of the two legs of the right triangle are found by solving a quadratic equation derived from the theorem. The two legs are 20 cm and 15 cm long, respectively.
Step-by-step explanation:
The problem involves finding the lengths of the legs of a right triangle when the hypotenuse is given and one leg is 5 centimeters shorter than the other. We use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the legs (a and b) is equal to the square of the hypotenuse (c). The theorem is represented by the equation a² + b² = c².
Let the longer leg be x cm. Then, the shorter leg is (x - 5) cm. Since the hypotenuse is 25 cm, we can express this as:
(x - 5)² + x² = 25²
Solving for x:
x² + (x - 5)² = 625
x² + x² - 10x + 25 = 625
2x² - 10x - 600 = 0
Dividing everything by 2:
x² - 5x - 300 = 0
Solving this quadratic equation, we find that x = 20. Therefore, the lengths of the two legs are:
- Longer leg: 20 cm
- Shorter leg: 15 cm
The answer is options B) 15 cm and 20 cm.