Finding The Hypotenuse Of A Right Triangle

#Finding the hypotenuse of a right triangle can be a challenging task, but it becomes straightforward with the correct approach*. In this article, we will explore* a specific right triangle problem and guide you through the steps to arrive at the solution. We will break down the problem, discuss the relevant concepts, and provide a clear explanation of the solution process. Whether you are a student preparing for an exam or simply someone looking to refresh your geometry skills, this article will equip you with the knowledge to tackle similar problems with confidence.

Problem Statement

Let's delve into the problem at hand. The question presents us with a right triangle where the shortest side measures $3 \sqrt{3}$ inches. Additionally, we are given that one angle of the triangle measures $60^{\circ}$. The objective is to determine the length of the hypotenuse of this triangle, expressed in inches. The provided answer choices are:

• A. 6
• B. $6 \sqrt{2}$
• C. 3
• D. $6 \sqrt{3}$

Understanding the Concepts

Before we jump into the solution, it's essential to solidify our understanding of the underlying concepts. This problem hinges on our knowledge of right triangles, trigonometric ratios, and special right triangles.

Right Triangles

A right triangle is a triangle containing one angle that measures 90 degrees. This angle is known as the right angle, and the side opposite the right angle is called the hypotenuse. The hypotenuse is always the longest side of a right triangle. The other two sides are called legs. Understanding these fundamental properties of right triangles is crucial for solving geometric problems.

Trigonometric Ratios

Trigonometric ratios provide a powerful tool for relating the angles and sides of right triangles. The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). These ratios are defined as follows:

• Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
• Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
• Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

These ratios can be expressed mathematically as:

• sin(θ) = Opposite / Hypotenuse
• cos(θ) = Adjacent / Hypotenuse
• tan(θ) = Opposite / Adjacent

where θ represents the angle in question.

Special Right Triangles

Certain right triangles, known as special right triangles, have specific angle and side ratios that are worth memorizing. The two most common special right triangles are the 30-60-90 triangle and the 45-45-90 triangle. In a 30-60-90 triangle, the angles measure 30 degrees, 60 degrees, and 90 degrees. The sides of a 30-60-90 triangle are in the ratio 1 : √3 : 2, where the shortest side is opposite the 30-degree angle, the middle side is opposite the 60-degree angle, and the longest side (the hypotenuse) is opposite the 90-degree angle. Recognizing these special triangles can significantly simplify problem-solving.

Solution

Now that we have reviewed the necessary concepts, let's tackle the problem at hand. We are given a right triangle with a shortest side of $3 \sqrt{3}$ inches and one angle measuring $60^{\circ}$. Since it's a right triangle, we know one angle is $90^{\circ}$. With angles of $60^{\circ}$ and $90^{\circ}$, the remaining angle must be $30^{\circ}$ (because the angles in a triangle add up to $180^{\circ}$). This confirms that we are dealing with a 30-60-90 triangle.

Identifying the Sides

In a 30-60-90 triangle, the shortest side is opposite the 30-degree angle. We are given that the shortest side measures $3 \sqrt{3}$ inches. The sides of a 30-60-90 triangle are in the ratio 1 : √3 : 2. This means:

• Side opposite 30° angle (shortest side) : x
• Side opposite 60° angle : x√3
• Hypotenuse (opposite 90° angle) : 2x

Since the shortest side is $3 \sqrt{3}$, we can set x = $3 \sqrt{3}$.

Calculating the Hypotenuse

The hypotenuse is 2x, so we substitute the value of x:

Hypotenuse = 2 * $3 \sqrt{3}$ = $6 \sqrt{3}$ inches.

Therefore, the length of the hypotenuse of the triangle is $6 \sqrt{3}$ inches.

Selecting the Correct Answer

Looking at the answer choices, we find that option D, $6 \sqrt{3}$, matches our calculated hypotenuse length. Thus, the correct answer is D.

Why Other Options Are Incorrect

To reinforce our understanding, let's examine why the other answer choices are incorrect:

• A. 6: This value does not correspond to the correct ratio of sides in a 30-60-90 triangle given the shortest side is $3 \sqrt{3}$.
• B. $6 \sqrt{2}$: This value might arise from confusion with 45-45-90 triangles, where the sides are in the ratio 1:1:√2. However, our triangle is a 30-60-90 triangle.
• C. 3: This value is significantly smaller than what the hypotenuse should be, given the length of the shortest side. In a right triangle, the hypotenuse is always the longest side.

Key Takeaways

This problem highlights the importance of understanding special right triangles and trigonometric ratios. By recognizing the 30-60-90 triangle pattern and applying the side ratios, we efficiently determined the hypotenuse length. Key takeaways from this problem include:

• Recognizing Special Right Triangles: Identifying 30-60-90 and 45-45-90 triangles can simplify calculations.
• Understanding Trigonometric Ratios: Sine, cosine, and tangent relate angles and sides in right triangles.
• Applying Ratios: Using the ratios of sides in special right triangles allows for quick calculation of unknown side lengths.
• Logical Elimination: Understanding the properties of triangles helps eliminate incorrect answer choices.

Practice Problems

To further solidify your understanding, try solving these practice problems:

1. The longer leg of a 30-60-90 triangle measures 8 inches. What is the length of the hypotenuse?
2. In a 45-45-90 triangle, one leg measures 5 inches. What is the length of the hypotenuse?
3. A right triangle has a hypotenuse of 10 inches and one angle of 30 degrees. What is the length of the shortest side?

Conclusion

Mastering the concepts of right triangles, trigonometric ratios, and special right triangles is crucial for success in geometry and trigonometry. By understanding these principles and practicing problem-solving, you can confidently tackle a wide range of geometric challenges. Remember to carefully analyze the given information, identify relevant relationships, and apply the appropriate formulas and ratios to arrive at the correct solution. With consistent practice, you'll become proficient in solving right triangle problems and excel in your mathematical endeavors. Remember, practice makes perfect, and with each problem you solve, your understanding will deepen, and your confidence will grow. So, keep practicing, keep learning, and keep exploring the fascinating world of mathematics!