Special Right TrianglesThe Hypotenuse Of A $45 {\circ}-45 {\circ}-90^{\circ}$ Triangle Has A Length Of $12 \sqrt{6}$ Inches. What Is The Length Of Its Leg?Write Your Answer In Simplest Radical Form. □ \square □ Inches
Introduction
In the world of geometry, special right triangles are a fascinating topic that has captivated mathematicians for centuries. Among these special triangles, the 45-45-90 triangle stands out for its unique properties and applications. In this article, we will delve into the world of 45-45-90 triangles, exploring their characteristics, properties, and how to calculate their lengths. We will also apply this knowledge to solve a problem involving a 45-45-90 triangle with a given hypotenuse length.
What are Special Right Triangles?
Special right triangles are triangles with specific angle measures that result in unique side length ratios. These triangles are classified into three categories: 30-60-90, 45-45-90, and 60-60-90 triangles. Each category has its own set of properties and characteristics that make them useful in various mathematical and real-world applications.
Properties of 45-45-90 Triangles
A 45-45-90 triangle is a right triangle with two 45-degree angles and one 90-degree angle. The properties of a 45-45-90 triangle are as follows:
- The two legs (sides opposite the 45-degree angles) are equal in length.
- The hypotenuse (side opposite the 90-degree angle) is √2 times the length of each leg.
- The ratio of the lengths of the sides is 1:1:√2.
Calculating the Length of a Leg in a 45-45-90 Triangle
To calculate the length of a leg in a 45-45-90 triangle, we can use the following formula:
Leg Length = Hypotenuse Length / √2
This formula is derived from the properties of a 45-45-90 triangle, where the hypotenuse is √2 times the length of each leg.
Solving the Problem
Now, let's apply this knowledge to solve the problem stated in the introduction. We are given a 45-45-90 triangle with a hypotenuse length of 12√6 inches. We need to find the length of its leg.
Step 1: Identify the Given Information
- Hypotenuse length: 12√6 inches
Step 2: Apply the Formula
- Leg Length = Hypotenuse Length / √2
- Leg Length = (12√6) / √2
Step 3: Simplify the Expression
- To simplify the expression, we can multiply the numerator and denominator by √2 to eliminate the radical in the denominator.
- Leg Length = (12√6) / √2 * (√2 / √2)
- Leg Length = (12√6 * √2) / (√2 * √2)
- Leg Length = 12√12 / 2
- Leg Length = 6√12
Step 4: Simplify the Radical
- To simplify the radical, we can express √12 as √(4 * 3) = √4 * √3 = 2√3.
- Leg Length = 6 * 2√3
- Leg Length = 12√3
Therefore, the length of the leg of the 45-45-90 triangle is 12√3 inches.
Conclusion
In this article, we explored the properties and characteristics of 45-45-90 triangles, including their unique side length ratios and how to calculate the length of a leg. We applied this knowledge to solve a problem involving a 45-45-90 triangle with a given hypotenuse length. By understanding the properties of special right triangles, we can unlock the secrets of geometry and solve a wide range of mathematical problems.
Frequently Asked Questions
Q: What is a 45-45-90 triangle?
A: A 45-45-90 triangle is a right triangle with two 45-degree angles and one 90-degree angle.
Q: What are the properties of a 45-45-90 triangle?
A: The properties of a 45-45-90 triangle are as follows:
- The two legs (sides opposite the 45-degree angles) are equal in length.
- The hypotenuse (side opposite the 90-degree angle) is √2 times the length of each leg.
- The ratio of the lengths of the sides is 1:1:√2.
Q: How do I calculate the length of a leg in a 45-45-90 triangle?
A: To calculate the length of a leg in a 45-45-90 triangle, you can use the following formula:
Leg Length = Hypotenuse Length / √2
Q: What is the length of the leg of a 45-45-90 triangle with a hypotenuse length of 12√6 inches?
Q&A: Frequently Asked Questions about 45-45-90 Triangles
Q: What is a 45-45-90 triangle?
A: A 45-45-90 triangle is a right triangle with two 45-degree angles and one 90-degree angle. This type of triangle is also known as an isosceles right triangle, as the two legs (sides opposite the 45-degree angles) are equal in length.
Q: What are the properties of a 45-45-90 triangle?
A: The properties of a 45-45-90 triangle are as follows:
- The two legs (sides opposite the 45-degree angles) are equal in length.
- The hypotenuse (side opposite the 90-degree angle) is √2 times the length of each leg.
- The ratio of the lengths of the sides is 1:1:√2.
Q: How do I calculate the length of a leg in a 45-45-90 triangle?
A: To calculate the length of a leg in a 45-45-90 triangle, you can use the following formula:
Leg Length = Hypotenuse Length / √2
This formula is derived from the properties of a 45-45-90 triangle, where the hypotenuse is √2 times the length of each leg.
Q: What is the relationship between the hypotenuse and the legs of a 45-45-90 triangle?
A: The hypotenuse of a 45-45-90 triangle is √2 times the length of each leg. This means that if you know the length of the hypotenuse, you can calculate the length of each leg by dividing the hypotenuse length by √2.
Q: Can I use the Pythagorean theorem to calculate the length of a leg in a 45-45-90 triangle?
A: Yes, you can use the Pythagorean theorem to calculate the length of a leg in a 45-45-90 triangle. The Pythagorean theorem states that a^2 + b^2 = c^2, where a and b are the lengths of the legs and c is the length of the hypotenuse. In a 45-45-90 triangle, the two legs are equal in length, so you can use the formula a^2 + a^2 = c^2 to calculate the length of each leg.
Q: What is the simplest radical form of the length of a leg in a 45-45-90 triangle?
A: The simplest radical form of the length of a leg in a 45-45-90 triangle is √2 times the length of the hypotenuse divided by 2. This can be expressed as:
Leg Length = (Hypotenuse Length / √2) / 2
Q: Can I use a calculator to calculate the length of a leg in a 45-45-90 triangle?
A: Yes, you can use a calculator to calculate the length of a leg in a 45-45-90 triangle. Simply enter the length of the hypotenuse and divide it by √2 to get the length of each leg.
Q: What are some real-world applications of 45-45-90 triangles?
A: 45-45-90 triangles have many real-world applications, including:
- Building design and construction
- Engineering and architecture
- Art and design
- Physics and mathematics
Conclusion
In this article, we have explored the properties and characteristics of 45-45-90 triangles, including their unique side length ratios and how to calculate the length of a leg. We have also answered some frequently asked questions about 45-45-90 triangles, including how to calculate the length of a leg and what are some real-world applications of these triangles. By understanding the properties of special right triangles, we can unlock the secrets of geometry and solve a wide range of mathematical problems.
Additional Resources
Frequently Asked Questions
Q: What is a 45-45-90 triangle?
A: A 45-45-90 triangle is a right triangle with two 45-degree angles and one 90-degree angle.
Q: What are the properties of a 45-45-90 triangle?
A: The properties of a 45-45-90 triangle are as follows:
- The two legs (sides opposite the 45-degree angles) are equal in length.
- The hypotenuse (side opposite the 90-degree angle) is √2 times the length of each leg.
- The ratio of the lengths of the sides is 1:1:√2.
Q: How do I calculate the length of a leg in a 45-45-90 triangle?
A: To calculate the length of a leg in a 45-45-90 triangle, you can use the following formula:
Leg Length = Hypotenuse Length / √2
Q: What is the relationship between the hypotenuse and the legs of a 45-45-90 triangle?
A: The hypotenuse of a 45-45-90 triangle is √2 times the length of each leg. This means that if you know the length of the hypotenuse, you can calculate the length of each leg by dividing the hypotenuse length by √2.
Q: Can I use the Pythagorean theorem to calculate the length of a leg in a 45-45-90 triangle?
A: Yes, you can use the Pythagorean theorem to calculate the length of a leg in a 45-45-90 triangle. The Pythagorean theorem states that a^2 + b^2 = c^2, where a and b are the lengths of the legs and c is the length of the hypotenuse. In a 45-45-90 triangle, the two legs are equal in length, so you can use the formula a^2 + a^2 = c^2 to calculate the length of each leg.
Q: What is the simplest radical form of the length of a leg in a 45-45-90 triangle?
A: The simplest radical form of the length of a leg in a 45-45-90 triangle is √2 times the length of the hypotenuse divided by 2. This can be expressed as:
Leg Length = (Hypotenuse Length / √2) / 2
Q: Can I use a calculator to calculate the length of a leg in a 45-45-90 triangle?
A: Yes, you can use a calculator to calculate the length of a leg in a 45-45-90 triangle. Simply enter the length of the hypotenuse and divide it by √2 to get the length of each leg.
Q: What are some real-world applications of 45-45-90 triangles?
A: 45-45-90 triangles have many real-world applications, including:
- Building design and construction
- Engineering and architecture
- Art and design
- Physics and mathematics