Final answer:
Using the Pythagorean theorem, none of the given statements about the unknown length in the triangle are correct. The correct length of the unknown leg is approximately 10.25 meters, not 23 or 7 meters, as claimed in the statements, and neither equation provided is appropriate for finding the unknown leg in the context given.
Step-by-step explanation:
To determine which statements are true about the unknown length in the given triangle, we must use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two legs (a and b) is equal to the square of the hypotenuse (c): a² + b² = c². In this case, we know one leg is 8 meters, and the hypotenuse is 13 meters, but the length of the other leg (b) is unknown.
Let's address each statement:
- The equation (13)² + b² = (8)² is incorrect because it assumes that the hypotenuse is the unknown side and is not meant for finding the unknown leg (b). Thus, this statement is false.
- The equation (13)² + (8)² = c² is the correct application of the Pythagorean theorem for finding the hypotenuse when both legs are known, which is not the scenario we have. Therefore, this statement is also false since we are given the hypotenuse length.
- Claiming that the length of the unknown leg is 23 meters cannot be evaluated without calculation, so we need to perform the calculation: b = √((13)² - (8)²). After performing the calculations, b = √(169 - 64) = √105 ≈ 10.25 (not 23 meters), so this statement is false.
- The length of the unknown leg is not 7 meters, as it is approximately 10.25 meters after solving using the Pythagorean theorem. Therefore, this statement is false.
No statements regarding the length of the unknown leg are correct if the triangle has a hypotenuse of 13 meters and one leg of 8 meters.