30°-60°-90° Triangle Hypotenuse 24 Inches Leg Lengths

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Understanding the properties of special right triangles, particularly the 30°-60°-90° triangle, is crucial in geometry and trigonometry. These triangles possess specific side length ratios that allow us to determine unknown sides when given one side. In this article, we will explore how to calculate the lengths of the legs in a 30°-60°-90° triangle when the hypotenuse is known. Specifically, we will address the problem: If the hypotenuse of a 30°-60°-90° triangle measures 24 inches, which could be the length of a leg of the triangle? We will analyze the given options and use the 30°-60°-90° triangle theorem to determine the correct answers.

Understanding 30°-60°-90° Triangles

A 30°-60°-90° triangle is a special type of right triangle with angles measuring 30 degrees, 60 degrees, and 90 degrees. The sides of this triangle are in a specific ratio, which is fundamental to solving problems involving these triangles. This ratio is 1:√3:2, where:

  • The side opposite the 30° angle (the shorter leg) is represented by x.
  • The side opposite the 60° angle (the longer leg) is represented by x√3.
  • The side opposite the 90° angle (the hypotenuse) is represented by 2x.

This ratio is derived from the properties of equilateral triangles and the Pythagorean theorem. When an equilateral triangle is bisected, it forms two 30°-60°-90° triangles. The side lengths of the equilateral triangle become the hypotenuse and one leg of the 30°-60°-90° triangle, while the altitude bisecting the triangle becomes the other leg. This geometric relationship is the basis for the 1:√3:2 ratio.

Knowing this ratio, we can easily find the lengths of the other two sides if we are given the length of any one side. For instance, if we know the hypotenuse, we can find the shorter leg by dividing the hypotenuse by 2, and then find the longer leg by multiplying the shorter leg by √3. These relationships are critical in various applications, including architecture, engineering, and navigation, where 30°-60°-90° triangles frequently appear.

Solving the Problem: Hypotenuse = 24 Inches

In our problem, we are given that the hypotenuse of the 30°-60°-90° triangle measures 24 inches. To find the lengths of the legs, we use the ratio relationships we discussed earlier. Let's denote the shorter leg as x, the longer leg as x√3, and the hypotenuse as 2x. We know that:

2x = 24 inches

To find x, which is the length of the shorter leg, we divide both sides of the equation by 2:

x = 24 / 2 = 12 inches

So, the shorter leg of the triangle is 12 inches. This means option C, 12 inches, is a correct answer. Now, to find the length of the longer leg, we multiply the shorter leg by √3:

Longer leg = x√3 = 12√3 inches

Thus, the longer leg of the triangle is 12√3 inches. This confirms that option D, 12√3 inches, is also a correct answer.

Now, let's examine the remaining options to ensure they are not valid. Option A is 15 inches, and Option B is 9√3 inches. We already know the two leg lengths are 12 inches and 12√3 inches. Neither of these matches 15 inches or 9√3 inches. Therefore, options A and B are incorrect.

Detailed Analysis of the Options

To further solidify our understanding, let’s analyze each option in detail:

Option A: 15 inches

If the shorter leg were 15 inches, the hypotenuse would be 2 * 15 = 30 inches, which contradicts the given hypotenuse length of 24 inches. If the longer leg were 15 inches, then x√3 = 15, and x = 15 / √3. Multiplying the numerator and denominator by √3 to rationalize the denominator gives x = (15√3) / 3 = 5√3. In this case, the hypotenuse would be 2 * 5√3 = 10√3 inches, which again contradicts the given length of 24 inches. Therefore, 15 inches cannot be the length of a leg in this 30°-60°-90° triangle.

Option B: 9√3 inches

If the shorter leg were 9√3 inches, the hypotenuse would be 2 * 9√3 = 18√3 inches, which is not equal to 24 inches. If the longer leg were 9√3 inches, then x√3 = 9√3, so x = 9 inches. In this case, the hypotenuse would be 2 * 9 = 18 inches, which also does not match the given hypotenuse of 24 inches. Hence, 9√3 inches is not a possible leg length for this triangle.

Option C: 12 inches

As we calculated earlier, if the hypotenuse is 24 inches, the shorter leg (x) is indeed 12 inches. This option aligns perfectly with the properties of a 30°-60°-90° triangle and is therefore a correct answer.

Option D: 12√3 inches

The longer leg is found by multiplying the shorter leg (12 inches) by √3, which gives us 12√3 inches. This matches our calculated length for the longer leg, making this option a correct answer as well.

Conclusion

In summary, given that the hypotenuse of a 30°-60°-90° triangle is 24 inches, the possible lengths of a leg of the triangle are 12 inches and 12√3 inches. Options C and D are the correct answers. This problem illustrates the importance of understanding the side ratios in special right triangles and how to apply these ratios to solve for unknown side lengths. By mastering these concepts, one can tackle a wide range of geometry and trigonometry problems with confidence.

Understanding the 30°-60°-90° triangle ratios not only helps in solving mathematical problems but also provides a foundation for more advanced concepts in trigonometry and calculus. The consistent application of these ratios in various scenarios reinforces their importance in practical applications and theoretical mathematics alike.

The ability to quickly identify and apply these ratios can also be beneficial in standardized tests and competitive exams, where time efficiency is crucial. By practicing with different variations of these problems, students can develop a strong intuition for the relationships within 30°-60°-90° triangles and improve their overall problem-solving skills in mathematics.

In conclusion, the lengths of the legs of a 30°-60°-90° triangle with a hypotenuse of 24 inches are 12 inches and 12√3 inches. The correct options are C and D, which demonstrate a solid understanding of the fundamental properties of these special right triangles.