The Hypotenuse Of A 45-45-90 Triangle Measures 128 Cm

Decoding the 45-45-90 Triangle: When the Hypotenuse Measures 128 cm

Understanding the properties of a 45-45-90 triangle, also known as an isosceles right triangle, is fundamental in geometry and trigonometry. This special right triangle, characterized by its two congruent legs and a right angle, pops up frequently in various applications, from architectural design to surveying. This article delves deep into the characteristics of this triangle, focusing specifically on a scenario where the hypotenuse measures 128 cm. We'll explore how to determine the lengths of the legs, discuss the underlying mathematical principles, and even touch upon practical applications of this knowledge.

Understanding the 45-45-90 Triangle

Before we jump into the specifics of our 128 cm hypotenuse, let's solidify our understanding of the 45-45-90 triangle. The name itself gives us crucial clues:

• 45-45-90: This refers to the three angles within the triangle. Two angles measure 45 degrees each, and the third angle is a right angle (90 degrees). The sum of angles in any triangle always equals 180 degrees, and this triangle perfectly exemplifies that rule.
• Isosceles Right Triangle: Because two angles are equal (45 degrees each), the sides opposite these angles are also equal in length. This is a defining characteristic of an isosceles triangle. The "right triangle" part simply signifies the presence of the 90-degree angle.

This unique combination of angle and side properties leads to a simplified ratio for the sides. If we let 'x' represent the length of each leg, then the hypotenuse (the side opposite the 90-degree angle) will always be x√2. This relationship is crucial for solving problems involving 45-45-90 triangles.

Calculating the Legs: A Step-by-Step Approach

Now, let's tackle our specific problem: a 45-45-90 triangle with a hypotenuse of 128 cm. Our goal is to find the length of each leg.

Step 1: Understand the Relationship

Remember the fundamental relationship: hypotenuse = leg × √2

Step 2: Set Up the Equation

We know the hypotenuse (128 cm), so we can set up the equation:

128 cm = x√2

where 'x' represents the length of each leg.

Step 3: Solve for x

To solve for 'x', we need to isolate it on one side of the equation. We can do this by dividing both sides by √2:

x = 128 cm / √2

Step 4: Rationalize the Denominator (Optional but Recommended)

While the above equation is correct, it's generally preferred to have a rational number (a number without a square root in the denominator). To achieve this, we multiply both the numerator and the denominator by √2:

x = (128 cm × √2) / (√2 × √2)

x = (128√2 cm) / 2

x = 64√2 cm

Step 5: Approximate the Value (If Necessary)

The exact length of each leg is 64√2 cm. However, for practical applications, you might need an approximate decimal value. Using a calculator, we find that:

√2 ≈ 1.414

Therefore:

x ≈ 64 cm × 1.414

x ≈ 90.496 cm

So, each leg of the 45-45-90 triangle with a hypotenuse of 128 cm is approximately 90.5 cm.

The Pythagorean Theorem: A Deeper Dive

The solution above utilizes the inherent properties of a 45-45-90 triangle. However, we can also solve this using the Pythagorean Theorem, a cornerstone of geometry. The theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In our case:

hypotenuse² = leg² + leg²

Since the legs are equal in length in a 45-45-90 triangle, we can simplify this to:

hypotenuse² = 2 × leg²

Substituting the known hypotenuse length:

(128 cm)² = 2 × leg²

16384 cm² = 2 × leg²

leg² = 8192 cm²

leg = √8192 cm

leg ≈ 90.5 cm

This confirms our previous calculation using the simplified ratio. The Pythagorean Theorem provides a more general approach applicable to any right-angled triangle, not just 45-45-90 triangles.

Practical Applications of 45-45-90 Triangles

The 45-45-90 triangle, with its simple yet powerful properties, finds its way into many real-world scenarios:

• Construction and Architecture: This triangle is often used in construction to ensure precise angles and measurements. For example, the diagonal bracing in square structures often forms 45-45-90 triangles.
• Surveying and Mapping: Surveyors use the properties of this triangle to calculate distances and heights indirectly, making it a valuable tool in land measurement and mapping.
• Computer Graphics and Game Design: The simplicity of its ratios makes it ideal for creating various shapes and designs in 2D and 3D environments.
• Navigation: Understanding 45-45-90 triangles can be helpful in calculating distances and bearings in navigation.

Trigonometric Functions and the 45-45-90 Triangle

Trigonometric functions—sine, cosine, and tangent—are crucial tools in geometry. In a 45-45-90 triangle, these functions have particularly simple values:

• sin(45°) = cos(45°) = 1/√2 = √2/2 ≈ 0.707
• tan(45°) = 1

These values are derived directly from the ratio of sides in the triangle. Understanding these values is crucial for solving more complex trigonometry problems involving 45-45-90 triangles.

Frequently Asked Questions (FAQs)

Q: Why is the ratio of sides in a 45-45-90 triangle always x:x:x√2?

A: This ratio stems directly from the Pythagorean theorem and the fact that the two legs are equal in length. The equal legs create an isosceles right triangle with inherent symmetrical properties resulting in this specific ratio.

Q: Can I use the Pythagorean theorem to solve for the legs even if I know the hypotenuse?

A: Absolutely! The Pythagorean theorem is a universal tool for solving right-angled triangles. It provides a robust method regardless of which sides you know.

Q: What if I have a 45-45-90 triangle where only one leg is given?

A: If you know one leg (let's say 'x'), then you immediately know the other leg is also 'x', and the hypotenuse is x√2. The problem becomes even simpler.

Q: Are there other special right triangles besides the 45-45-90 triangle?

A: Yes! The 30-60-90 triangle is another special right triangle with its own unique side ratio (x: x√3 : 2x). Understanding both of these special triangles simplifies various geometric calculations.

Conclusion: Mastering the 45-45-90 Triangle

The 45-45-90 triangle, with its inherent simplicity and widespread applicability, remains a fundamental concept in geometry and related fields. By understanding its properties, particularly the simplified side ratio and its connection to the Pythagorean Theorem, you can effectively solve a wide range of problems involving this special right triangle. The example of a 128 cm hypotenuse showcases how to determine the lengths of the legs using both the simplified ratio and the Pythagorean Theorem, underscoring the versatility and power of geometric principles in solving real-world problems. Remember, mastering these concepts builds a strong foundation for tackling more advanced topics in mathematics and beyond.