The hypotenuse of a right triangle measures 13 cm and the difference of the legs is 7 cm. How many centimeters does each leg measure?

Let's call the lengths of the legs of the right triangle "a" and "b," and the hypotenuse "c." According to the Pythagorean theorem, the relationship between the lengths of the legs and the hypotenuse in a right triangle is: c² = a² + b² In this case, we're given that the hypotenuse "c" measures 13 cm, and the difference of the legs is 7 cm, which means a - b = 7. We can set up a system of equations to solve for a and b: c² = a² + b² (Pythagorean theorem) a - b = 7 (difference of the legs) Let's solve this system of equations: From equation (2), we can express "a" in terms of "b": a = b + 7 Now, substitute this expression for "a" into equation (1): c² = (b + 7)² + b² Expand the expression: c² = b² + 14b + 49 + b² Combine like terms: c² = 2b² + 14b + 49 Now, we know that c² is 13² because the hypotenuse measures 13 cm: 169 = 2b² + 14b + 49 Subtract 169 from both sides: 0 = 2b² + 14b + 49 - 169 0 = 2b² + 14b - 120 Divide the entire equation by 2 to simplify: 0 = b² + 7b - 60 Now, let's solve this quadratic equation for "b" by factoring: 0 = (b + 12)(b - 5) Set each factor equal to zero: b + 12 = 0 b = -12 b - 5 = 0 b = 5 Now, we have two possible values for "b": -12 and 5. However, since lengths cannot be negative in this context, we discard the negative value, and b = 5 cm. Now that we have found the value of "b," we can find the value of "a" using the equation a = b + 7: a = 5 + 7 a = 12 cm So, one leg of the right triangle measures 5 cm, and the other leg measures 12 cm.