Final answer:
Both legs of a right triangle with one angle of 45° and a hypotenuse of 312 inches are approximately 220.5 inches long, calculated using the properties of a 45°-45°-90° triangle with three significant figures.
Step-by-step explanation:
To calculate the side lengths of a right triangle with one angle of 45° and a hypotenuse of 312 inches, we can use the properties of a 45°-45°-90° triangle, where the legs are equal in length, and the hypotenuse is \(\sqrt{2}\) times longer than each leg. We can set one of the legs to 'x' and write the equation as \(x^2 + x^2 = 312^2\).
Combining the like terms, we get \(2x^2 = 312^2\). Dividing both sides by 2 yields \(x^2 = \frac{312^2}{2}\).
Taking the square root of both sides gives us the length of one leg, \(x = \sqrt{\frac{312^2}{2}}\). Therefore, both legs of the triangle have a length of \(\sqrt{\frac{312^2}{2}} \approx 220.5\) inches when calculated using three significant figures.