7. What is the length of the unknown leg of a right triangle that has one leg measuring 8 feet and a hypotenuse of 12 feet? (Round to the nearest tenth)
Answer: D
The length of the unknown leg is 8.9 feet.
To find the length of the unknown leg in a right triangle with one leg measuring 8 feet and a hypotenuse of 12 feet, we apply the Pythagorean theorem. By calculating the square root of the difference between the square of the hypotenuse and the square of the known leg, we arrive at approximately 8.9 feet.
A) 40 feet
This option is incorrect as it greatly exceeds the possible length of the unknown leg based on the Pythagorean theorem. The maximum length of any leg in a right triangle cannot exceed the length of the hypotenuse.
B) 4 feet
While this option is a plausible length, it does not satisfy the Pythagorean theorem when calculated with the given leg and hypotenuse. The calculation shows that the length must be greater than 4 feet.
C) 14.4 feet
This option is also incorrect because it exceeds the length of the hypotenuse. In a right triangle, the length of each leg must always be less than the hypotenuse.
D) 8.9 feet
This option is correct. By applying the Pythagorean theorem, we calculate the unknown leg as follows: \( c^2 = a^2 + b^2 \) leads us to \( 12^2 = 8^2 + b^2 \). Solving yields \( b \approx 8.9 \) feet.
Conclusion
The correct answer is definitively 8.9 feet, as it satisfies the Pythagorean theorem when applied to the given dimensions of the triangle. All other options either exceed the maximum possible leg length or do not align with the mathematical requirements of the right triangle, thus confirming that D is the only viable solution.