Final answer:
Using the Pythagorean theorem, the lengths of the legs of the right triangle were calculated as approximately 52 inches and 49 inches.
Step-by-step explanation:
To solve the mathematical problem completely involving the lengths of the legs of a right triangle, we will use the Pythagorean theorem. We define the lengths of the legs as 'a' and 'b', where 'b = a - 3' since one leg is 3 inches shorter than the other. Given that the hypotenuse 'c' is 65 inches, we can write the equation as:
a² + (a - 3)² = 65²
Expanding the equation gives us:
a² + a² - 6a + 9 = 4225
Combining like terms results in:
2a² - 6a + 9 = 4225
Subtracting 4225 from both sides to set the equation to zero:
2a² - 6a - 4216 = 0
Dividing the entire equation by 2 simplifies it to:
a² - 3a - 2108 = 0
Solving this quadratic equation gives us two possible values for 'a', but only the positive value makes sense in this context. The positive solution for 'a' and the corresponding value of 'b = a - 3' gives us the lengths of the legs of the triangle. Calculating these values, we find that:
a ≈ 52 inches (approximate value)
b = 52 - 3
b ≈ 49 inches (approximate value)
Therefore, the complete calculated answer for the lengths of the legs of the right triangle are approximately 52 inches and 49 inches.