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The Hypotenuse of a 45°-45°-90° Triangle: Understanding the Relationship Between Legs and Hypotenuse
In the realm of geometry, triangles are a fundamental concept that has been studied extensively. Among the various types of triangles, the 45°-45°-90° triangle is a special case that has unique properties. In this article, we will delve into the relationship between the legs and the hypotenuse of a 45°-45°-90° triangle, and explore how to calculate the length of one of the legs when the hypotenuse is known.
What is a 45°-45°-90° Triangle?
A 45°-45°-90° triangle is a right-angled triangle with two angles measuring 45° each, and the third angle measuring 90°. This type of triangle has several distinct properties that make it useful in various mathematical and real-world applications. One of the key characteristics of a 45°-45°-90° triangle is that the two legs are equal in length, and the hypotenuse is √2 times the length of each leg.
The Relationship Between Legs and Hypotenuse
In a 45°-45°-90° triangle, the relationship between the legs and the hypotenuse can be expressed using the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this can be expressed as:
c² = a² + b²
In the case of a 45°-45°-90° triangle, since the two legs are equal in length, we can let a = b. Therefore, the Pythagorean theorem can be simplified to:
c² = 2a²
Calculating the Length of One of the Legs
Now that we have established the relationship between the legs and the hypotenuse, we can use this information to calculate the length of one of the legs when the hypotenuse is known. Let's consider the problem presented in the question: the hypotenuse of a 45°-45°-90° triangle measures 24 inches. We want to find the length of one of the legs.
Using the relationship c² = 2a², we can substitute the value of c (24 inches) and solve for a:
(24)² = 2a² 576 = 2a² a² = 288 a = √288 a = √(144 × 2) a = 12√2
Therefore, the length of one of the legs of the triangle is 12√2 inches.
In conclusion, the relationship between the legs and the hypotenuse of a 45°-45°-90° triangle is a fundamental concept in geometry. By understanding this relationship, we can use the Pythagorean theorem to calculate the length of one of the legs when the hypotenuse is known. In this article, we have demonstrated how to apply this concept to a specific problem, and arrived at the correct solution. Whether you are a student, a teacher, or simply someone interested in mathematics, this article has provided a comprehensive overview of the hypotenuse of a 45°-45°-90° triangle.
- What is a 45°-45°-90° triangle?
- What is the relationship between the legs and the hypotenuse of a 45°-45°-90° triangle?
- How can I calculate the length of one of the legs when the hypotenuse is known?
- What is the Pythagorean theorem, and how is it used in geometry?
- "Geometry: A Comprehensive Introduction" by Dan Pedoe
- "Mathematics for the Nonmathematician" by Morris Kline
- "The Pythagorean Theorem: A History" by Eli Maor
- Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
- Legs: The two sides of a right-angled triangle that meet at a right angle.
- Pythagorean theorem: A mathematical formula that describes the relationship between the lengths of the sides of a right-angled triangle.
- 45°-45°-90° triangle: A right-angled triangle with two angles measuring 45° each, and the third angle measuring 90°.
Frequently Asked Questions: The Hypotenuse of a 45°-45°-90° Triangle ====================================================================
Q: What is a 45°-45°-90° triangle?
A: A 45°-45°-90° triangle is a right-angled triangle with two angles measuring 45° each, and the third angle measuring 90°. This type of triangle has several distinct properties that make it useful in various mathematical and real-world applications.
Q: What is the relationship between the legs and the hypotenuse of a 45°-45°-90° triangle?
A: In a 45°-45°-90° triangle, the relationship between the legs and the hypotenuse can be expressed using the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this can be expressed as:
c² = a² + b²
In the case of a 45°-45°-90° triangle, since the two legs are equal in length, we can let a = b. Therefore, the Pythagorean theorem can be simplified to:
c² = 2a²
Q: How can I calculate the length of one of the legs when the hypotenuse is known?
A: To calculate the length of one of the legs when the hypotenuse is known, you can use the relationship c² = 2a². Let's consider the problem presented in the question: the hypotenuse of a 45°-45°-90° triangle measures 24 inches. We want to find the length of one of the legs.
Using the relationship c² = 2a², we can substitute the value of c (24 inches) and solve for a:
(24)² = 2a² 576 = 2a² a² = 288 a = √288 a = √(144 × 2) a = 12√2
Therefore, the length of one of the legs of the triangle is 12√2 inches.
Q: What is the Pythagorean theorem, and how is it used in geometry?
A: The Pythagorean theorem is a mathematical formula that describes the relationship between the lengths of the sides of a right-angled triangle. It states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this can be expressed as:
c² = a² + b²
The Pythagorean theorem is used extensively in geometry to calculate the lengths of the sides of right-angled triangles.
Q: What are some real-world applications of 45°-45°-90° triangles?
A: 45°-45°-90° triangles have several real-world applications, including:
- Construction: 45°-45°-90° triangles are used in construction to calculate the lengths of the sides of buildings and bridges.
- Engineering: 45°-45°-90° triangles are used in engineering to calculate the stresses and strains on materials.
- Design: 45°-45°-90° triangles are used in design to create symmetrical and aesthetically pleasing shapes.
Q: How can I use 45°-45°-90° triangles in my daily life?
A: 45°-45°-90° triangles can be used in various ways in your daily life, including:
- Measuring rooms: 45°-45°-90° triangles can be used to measure the lengths of the sides of rooms.
- Designing furniture: 45°-45°-90° triangles can be used to design symmetrical and aesthetically pleasing furniture.
- Calculating distances: 45°-45°-90° triangles can be used to calculate the distances between objects.
Q: What are some common mistakes to avoid when working with 45°-45°-90° triangles?
A: Some common mistakes to avoid when working with 45°-45°-90° triangles include:
- Confusing the legs and the hypotenuse: Make sure to identify the legs and the hypotenuse correctly.
- Using the wrong formula: Use the correct formula to calculate the lengths of the sides of the triangle.
- Not checking your work: Double-check your work to ensure that you have calculated the lengths of the sides correctly.
Q: How can I practice working with 45°-45°-90° triangles?
A: You can practice working with 45°-45°-90° triangles by:
- Solving problems: Practice solving problems involving 45°-45°-90° triangles.
- Creating diagrams: Create diagrams of 45°-45°-90° triangles to visualize the relationships between the sides.
- Using online resources: Use online resources, such as calculators and worksheets, to practice working with 45°-45°-90° triangles.