Select The Correct Answer.The Shortest Side Of A Right Triangle Measures $3 \sqrt{3}$ Inches. One Angle Of The Triangle Measures $60^{\circ}$. What Is The Length, In Inches, Of The Hypotenuse Of The Triangle?A. $6

Introduction

Right triangles are a fundamental concept in geometry, and understanding how to solve them is crucial for various mathematical and real-world applications. In this article, we will focus on solving right triangles with a given angle and side length. We will use the given information to find the length of the hypotenuse of a right triangle with a 60-degree angle and a side length of $3\sqrt{3}$ inches.

Understanding Right Triangles

A right triangle is a triangle with one angle that measures 90 degrees. The side opposite the 90-degree angle is called the hypotenuse, and the other two sides are called the legs. In a right triangle, the Pythagorean theorem states that the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

The 30-60-90 Triangle

The given triangle is a 30-60-90 triangle, which is a special type of right triangle with angles measuring 30, 60, and 90 degrees. In a 30-60-90 triangle, the side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is $sqrt{3}$ times the length of the side opposite the 30-degree angle.

Finding the Length of the Hypotenuse

Given that the shortest side of the triangle measures $3\sqrt{3}$ inches and one angle measures 60 degrees, we can use the properties of the 30-60-90 triangle to find the length of the hypotenuse. Since the side opposite the 60-degree angle is $sqrt{3}$ times the length of the side opposite the 30-degree angle, we can set up the following equation:

$$\frac{3\sqrt{3}}{\sqrt{3}} = x$$

where $x$ is the length of the hypotenuse.

Solving for x

To solve for $x$, we can simplify the equation by canceling out the $\sqrt{3}$ terms:

$$3 = x$$

However, this is not the correct answer. We need to find the length of the hypotenuse, which is twice the length of the side opposite the 30-degree angle. Therefore, we can multiply the length of the side opposite the 30-degree angle by 2:

$$x = 2 \times 3\sqrt{3}$$

Simplifying the Expression

To simplify the expression, we can multiply the terms:

$$x = 6\sqrt{3}$$

Conclusion

In this article, we used the properties of the 30-60-90 triangle to find the length of the hypotenuse of a right triangle with a 60-degree angle and a side length of $3\sqrt{3}$ inches. We set up an equation using the properties of the triangle and solved for the length of the hypotenuse. The final answer is $6\sqrt{3}$ inches.

Final Answer

Introduction

In our previous article, we discussed how to solve right triangles with a given angle and side length. We used the properties of the 30-60-90 triangle to find the length of the hypotenuse of a right triangle with a 60-degree angle and a side length of $3\sqrt{3}$ inches. In this article, we will provide a Q&A guide to help you understand and apply the concepts of solving right triangles.

Q: What is a right triangle?

A: A right triangle is a triangle with one angle that measures 90 degrees. The side opposite the 90-degree angle is called the hypotenuse, and the other two sides are called the legs.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem states that the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse. Mathematically, it can be expressed as:

$$a^2 + b^2 = c^2$$

where $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse.

Q: What is a 30-60-90 triangle?

A: A 30-60-90 triangle is a special type of right triangle with angles measuring 30, 60, and 90 degrees. In a 30-60-90 triangle, the side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is $sqrt{3}$ times the length of the side opposite the 30-degree angle.

Q: How do I find the length of the hypotenuse of a 30-60-90 triangle?

A: To find the length of the hypotenuse of a 30-60-90 triangle, you can use the following formula:

$$c = 2a$$

where $a$ is the length of the side opposite the 30-degree angle, and $c$ is the length of the hypotenuse.

Q: What is the relationship between the side opposite the 60-degree angle and the side opposite the 30-degree angle?

A: In a 30-60-90 triangle, the side opposite the 60-degree angle is $sqrt{3}$ times the length of the side opposite the 30-degree angle.

Q: How do I use the properties of the 30-60-90 triangle to solve a right triangle?

A: To use the properties of the 30-60-90 triangle to solve a right triangle, you can follow these steps:

1. Identify the type of triangle (30-60-90 or not).
2. Use the properties of the triangle to find the length of the hypotenuse or the side opposite the 60-degree angle.
3. Use the Pythagorean theorem to find the length of the other leg.

Q: What are some common mistakes to avoid when solving right triangles?

A: Some common mistakes to avoid when solving right triangles include:

• Not identifying the type of triangle.
• Not using the properties of the triangle correctly.
• Not using the Pythagorean theorem correctly.
• Not checking the units of the answer.

Conclusion

In this article, we provided a Q&A guide to help you understand and apply the concepts of solving right triangles. We discussed the properties of the 30-60-90 triangle and how to use them to solve right triangles. We also provided some common mistakes to avoid when solving right triangles. By following these guidelines, you can become more confident and proficient in solving right triangles.

Final Tips

• Practice, practice, practice: The more you practice solving right triangles, the more comfortable you will become with the concepts and formulas.
• Use visual aids: Visual aids such as diagrams and graphs can help you understand the concepts and relationships between the sides of the triangle.
• Check your work: Always check your work to ensure that you have used the correct formulas and units.

Final Answer

The final answer is: $\boxed{6\sqrt{3}}$