Ratio Of Longer Leg To Hypotenuse In A 30-60-90 Triangle

When delving into the fascinating world of geometry, certain special right triangles emerge as fundamental building blocks. Among these, the 30-60-90 triangle holds a place of prominence due to its predictable side ratios, making it a valuable tool for problem-solving in various mathematical contexts. This article aims to explore these ratios, specifically focusing on the relationship between the longer leg and the hypotenuse in a 30-60-90 triangle. We will dissect the underlying principles, derive the key ratios, and apply this knowledge to determine which of the provided options accurately represent the ratio of the longer leg to the hypotenuse.

A 30-60-90 triangle is a right triangle characterized by its angles measuring 30 degrees, 60 degrees, and 90 degrees. These triangles exhibit a consistent proportionality between their sides, a property that simplifies calculations and enables efficient problem-solving. To fully grasp the side ratios, it is essential to understand their derivation. We can begin by considering an equilateral triangle, a triangle with all three sides equal and all three angles measuring 60 degrees. If we bisect this equilateral triangle by drawing an altitude from one vertex to the midpoint of the opposite side, we create two congruent 30-60-90 triangles. This construction provides a visual and geometric foundation for understanding the side relationships.

In this newly formed 30-60-90 triangle, the hypotenuse corresponds to one of the original sides of the equilateral triangle. Let's denote the length of the hypotenuse as 2x. The shorter leg, opposite the 30-degree angle, is half the length of the hypotenuse, making it x. The longer leg, opposite the 60-degree angle, can be found using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Applying this to our triangle, we have: (2x)² = x² + (longer leg)². Solving for the longer leg, we get √( (2x)² - x²) = √(4x² - x²) = √3x. Thus, the sides of a 30-60-90 triangle are in the ratio x : √3x : 2x, corresponding to the shorter leg, longer leg, and hypotenuse, respectively.

Now that we have established the side ratios of a 30-60-90 triangle, we can focus specifically on the ratio of the longer leg to the hypotenuse. As derived earlier, the longer leg has a length of √3x, and the hypotenuse has a length of 2x. Therefore, the ratio of the longer leg to the hypotenuse is (√3x) / (2x). The x terms cancel out, leaving us with the ratio √3 : 2. This fundamental ratio is consistent for all 30-60-90 triangles, regardless of their size. This consistency arises from the inherent geometric properties of these triangles, ensuring that the proportional relationships between their sides remain constant.

The ratio √3 : 2 signifies that for every 2 units of length in the hypotenuse, the longer leg will measure √3 units. This ratio is crucial for solving a variety of problems involving 30-60-90 triangles, such as finding missing side lengths, determining areas, and analyzing geometric figures. Recognizing and applying this ratio allows for efficient and accurate calculations, saving valuable time and effort in mathematical problem-solving. Furthermore, understanding the derivation of this ratio, as discussed earlier, reinforces the underlying principles and enhances one's geometric intuition.

Having established that the ratio of the longer leg to the hypotenuse in a 30-60-90 triangle is √3 : 2, we can now evaluate the provided options to determine which ones are correct. The options are:

A. √2 : √3 B. √3 : 2 C. √3 : 1

Let's examine each option in detail:

Option A: √2 : √3 This ratio suggests that for every √3 units of length in the hypotenuse, the longer leg would measure √2 units. This does not align with the derived ratio of √3 : 2. The value √2 is approximately 1.414, while √3 is approximately 1.732. If we were to consider a triangle with a hypotenuse of √3 units, the longer leg, according to this ratio, would be shorter than the shorter leg, which is inconsistent with the properties of a 30-60-90 triangle. Therefore, Option A is incorrect.

Option B: √3 : 2 This ratio precisely matches the ratio we derived for the longer leg to the hypotenuse in a 30-60-90 triangle. For every 2 units of length in the hypotenuse, the longer leg measures √3 units. This ratio is consistent with the proportional relationships inherent in 30-60-90 triangles and is a fundamental characteristic of these triangles. Therefore, Option B is correct.

Option C: √3 : 1 This ratio implies that the longer leg is √3 times the length of the hypotenuse, which is not possible in any right triangle. The hypotenuse is always the longest side in a right triangle, and therefore, the ratio of the longer leg to the hypotenuse must be less than 1. The given ratio suggests that the longer leg is longer than the hypotenuse, contradicting the fundamental properties of a right triangle. Therefore, Option C is incorrect.

In conclusion, the ratio of the length of the longer leg to the length of the hypotenuse in a 30-60-90 triangle is √3 : 2. This ratio is a direct consequence of the triangle's unique angle measures and the resulting proportionality between its sides. By understanding the derivation of this ratio and applying it to various problems, one can efficiently solve geometric challenges involving 30-60-90 triangles. Among the given options, only Option B, √3 : 2, accurately represents this ratio. A deep understanding of these special right triangles and their ratios is invaluable for success in geometry and related mathematical disciplines.

Only option B is correct.

The correct answer is B. √3 : 2.