The Correct Answer and Explanation is:
Based on the mathematical principles governing right triangles, the correct choices from the options that are clearly visible are B (√3 / 2), E (√7 / 3), and F (√8 / 3).
Here is a detailed explanation of the reasoning.
In any right triangle, the hypotenuse is always the longest side. This fundamental rule means that the ratio of the length of any leg to the length of the hypotenuse must be a value less than 1. Any option that is equal to or greater than 1 can be immediately eliminated.
The question adds a second condition: we are looking for the ratio of the longer leg to the hypotenuse. Let’s analyze this mathematically. Let the legs of the right triangle be a and b, and let the hypotenuse be c. The Pythagorean theorem states that a² + b² = c². If we assume b is the longer leg, then b ≥ a.
To find the minimum possible value for this ratio, consider the case where the legs are equal, a = b. This is a 45-45-90 triangle. In this case, a² + a² = c², which simplifies to 2a² = c². The ratio of a leg to the hypotenuse is a/c. From our equation, a²/c² = 1/2, so a/c = √(1/2) = 1/√2 = √2 / 2.
When one leg is longer than the other, its ratio to the hypotenuse must be greater than this value. Therefore, the ratio R of the longer leg to the hypotenuse must fall within the range: √2 / 2 ≤ R < 1. In decimal form, this is approximately 0.707 ≤ R < 1.
Now we can evaluate the options:
- A: √2 ≈ 1.414. This is greater than 1, so it is incorrect.
- B: √3 / 2 ≈ 0.866. This value is between 0.707 and 1, so it is a valid ratio.
- E: √7 / 3 ≈ 2.646 / 3 ≈ 0.882. This value is also within the valid range.
- F: √8 / 3 ≈ 2.828 / 3 ≈ 0.943. This value is also within the valid range.
Therefore, options B, E, and F all represent mathematically possible ratios for the longer leg of a right triangle to its hypotenuse.
