The Hypotenuse Of A 45 45 90 Triangle Measures In

Decoding the 45-45-90 Triangle: Understanding its Hypotenuse

The 45-45-90 triangle, also known as an isosceles right triangle, is a special type of right-angled triangle that holds a unique and elegant relationship between its sides and angles. Understanding this relationship, particularly how to calculate the hypotenuse's length, is fundamental in various fields, from basic geometry to advanced trigonometry and even practical applications in construction and engineering. This comprehensive guide will delve deep into the properties of a 45-45-90 triangle, focusing specifically on how to determine the length of its hypotenuse given the length of one of its legs. We'll explore the underlying mathematical principles, provide step-by-step solutions, and address frequently asked questions to solidify your understanding.

Understanding the 45-45-90 Triangle's Properties

Before we jump into calculating the hypotenuse, let's establish a firm understanding of the 45-45-90 triangle's defining characteristics. The name itself reveals its key features:

• Two 45° Angles: The triangle possesses two angles measuring 45 degrees each. This is what makes it an isosceles triangle – meaning two of its sides are equal in length.
• One 90° Angle: As a right-angled triangle, it contains one right angle (90 degrees).
• Isosceles Property: The two legs (sides opposite the 45° angles) are congruent (equal in length). This is a direct consequence of the equal angles.

This inherent symmetry simplifies calculations significantly. The relationship between the legs and the hypotenuse follows a consistent pattern, which we'll explore in detail.

Calculating the Hypotenuse: The Pythagorean Theorem

The cornerstone of calculating the hypotenuse of any right-angled triangle, including our 45-45-90 triangle, is the Pythagorean Theorem. This fundamental theorem states that in a right-angled triangle, the square of the hypotenuse's length is equal to the sum of the squares of the lengths of the other two sides (legs). Mathematically, it's represented as:

a² + b² = c²

Where:

• a and b are the lengths of the two legs.
• c is the length of the hypotenuse.

Applying the Pythagorean Theorem to the 45-45-90 Triangle

Since the legs of a 45-45-90 triangle are equal (let's call their length 'x'), the Pythagorean Theorem simplifies to:

x² + x² = c²

This simplifies further to:

2x² = c²

To find the length of the hypotenuse (c), we take the square root of both sides:

c = x√2

This is the key formula for determining the hypotenuse of a 45-45-90 triangle. It shows that the hypotenuse's length is always √2 times the length of one of its legs.

Step-by-Step Calculation Examples

Let's illustrate this with a few examples:

Example 1:

A 45-45-90 triangle has legs of length 5 cm each. What is the length of its hypotenuse?

• Step 1: Identify the length of the legs (x). In this case, x = 5 cm.
• Step 2: Apply the formula: c = x√2
• Step 3: Substitute the value of x: c = 5√2 cm
• Step 4: Calculate the approximate value: c ≈ 7.07 cm

Therefore, the hypotenuse is approximately 7.07 cm long.

Example 2:

The hypotenuse of a 45-45-90 triangle measures 10 inches. What is the length of each leg?

• Step 1: We know c = 10 inches. We need to find x.
• Step 2: Use the formula: c = x√2
• Step 3: Substitute the value of c: 10 = x√2
• Step 4: Solve for x: x = 10 / √2
• Step 5: Rationalize the denominator (multiply both numerator and denominator by √2): x = (10√2) / 2
• Step 6: Simplify: x = 5√2 inches
• Step 7: Calculate the approximate value: x ≈ 7.07 inches

Therefore, each leg measures approximately 7.07 inches.

Example 3: Real-world Application

Imagine you're building a square patio. You want to place a diagonal support beam across the patio to add extra stability. If each side of the square patio is 8 feet, the diagonal beam forms the hypotenuse of a 45-45-90 triangle. How long should the support beam be?

• Step 1: The sides of the square (and therefore the legs of the triangle) are 8 feet each (x = 8 feet).
• Step 2: Use the formula: c = x√2
• Step 3: Substitute x: c = 8√2 feet
• Step 4: Calculate the approximate value: c ≈ 11.31 feet

You'll need a support beam approximately 11.31 feet long.

Beyond the Basics: Trigonometric Functions

While the Pythagorean Theorem provides a direct method, trigonometric functions offer another pathway to solve for the hypotenuse of a 45-45-90 triangle. Specifically, the sine and cosine functions can be used.

Since the triangle is isosceles, both of the non-right angles are 45 degrees. Therefore:

• sin(45°) = opposite/hypotenuse = x/c
• cos(45°) = adjacent/hypotenuse = x/c

Since sin(45°) = cos(45°) = 1/√2, we can use either equation to derive the same relationship: c = x√2.

Frequently Asked Questions (FAQs)

Q1: Can I use this formula for any right-angled triangle?

A1: No. This specific formula, c = x√2, only applies to 45-45-90 triangles because it relies on the equal leg lengths. For other right-angled triangles, you must use the general Pythagorean Theorem: a² + b² = c².

Q2: What if I only know the hypotenuse's length?

A2: As shown in Example 2, if you know the hypotenuse (c), you can rearrange the formula c = x√2 to solve for the leg length (x): x = c/√2. Remember to rationalize the denominator.

Q3: Are there any practical applications of 45-45-90 triangles?

A3: Yes! They are frequently encountered in:

• Construction: Calculating diagonal measurements, such as the length of a diagonal brace for a square structure.
• Engineering: Designing symmetrical structures and calculating distances in various applications.
• Navigation: Determining distances and bearings using right-angled triangles.
• Computer Graphics: Representing and manipulating geometric shapes.

Q4: How do I accurately calculate √2?

A4: You can use a calculator to obtain an accurate decimal approximation of √2 (approximately 1.414). For hand calculations, you might use estimation or approximation techniques.

Conclusion

The 45-45-90 triangle, with its inherent symmetry, offers a simplified approach to understanding right-angled triangle calculations. Mastering the formula c = x√2 for calculating the hypotenuse provides a valuable tool in various mathematical and practical scenarios. Remember that this formula is a specific application of the broader Pythagorean theorem, and understanding both principles is crucial for comprehensive geometrical problem-solving. Through understanding the principles outlined here and practicing with diverse examples, you can confidently navigate the world of 45-45-90 triangles and their applications. Remember to always double-check your calculations and utilize the appropriate tools for accurate measurements in real-world applications.