45 45 90 Triangle Examples With Answers

45-45-90 Triangle Examples with Answers: A Comprehensive Guide

The 45-45-90 triangle, also known as an isosceles right triangle, is a special right triangle characterized by its angles: 45°, 45°, and 90°. Understanding its properties is crucial in various fields, including geometry, trigonometry, and engineering. This comprehensive guide will delve into the characteristics of 45-45-90 triangles, explore numerous examples with detailed solutions, and provide practical applications to solidify your understanding.

Understanding the 45-45-90 Triangle

A 45-45-90 triangle is formed by bisecting a square along its diagonal. This creates two congruent right-angled triangles, each with two equal angles (45°) and a right angle (90°). This inherent symmetry leads to a unique relationship between its sides.

Key Properties:

• Two equal legs: The two legs (sides opposite the 45° angles) are always equal in length.
• Hypotenuse: The hypotenuse (the side opposite the 90° angle) is √2 times the length of each leg.

This relationship can be expressed as:

• Leg = x
• Leg = x
• Hypotenuse = x√2

Where 'x' represents the length of each leg.

Examples and Solutions

Let's explore various examples to illustrate the application of these properties. Each example will include a detailed step-by-step solution.

Example 1: Finding the Hypotenuse

Problem: A 45-45-90 triangle has legs of length 5 cm each. Find the length of the hypotenuse.

Solution:

1. Identify the known values: We know that both legs (a and b) have a length of 5 cm.
2. Apply the formula: Hypotenuse = Leg * √2
3. Substitute the values: Hypotenuse = 5 cm * √2
4. Calculate: Hypotenuse ≈ 7.07 cm

Therefore, the hypotenuse of the triangle is approximately 7.07 cm.

Example 2: Finding the Legs

Problem: A 45-45-90 triangle has a hypotenuse of length 10 inches. Find the length of each leg.

Solution:

1. Identify the known values: We know the hypotenuse is 10 inches.
2. Apply the formula (modified): Leg = Hypotenuse / √2
3. Substitute the values: Leg = 10 inches / √2
4. Rationalize the denominator: Leg = (10√2) / 2 inches
5. Simplify: Leg = 5√2 inches
6. Approximate: Leg ≈ 7.07 inches

Therefore, each leg of the triangle is approximately 7.07 inches.

Example 3: Real-world Application – Calculating Diagonal of a Square

Problem: A square garden has sides of 8 meters. What is the length of the diagonal path across the garden?

Solution:

1. Visualization: The diagonal of a square bisects it into two 45-45-90 triangles.
2. Identify the known values: The legs of each triangle are the sides of the square (8 meters each).
3. Apply the formula: Hypotenuse = Leg * √2
4. Substitute the values: Hypotenuse = 8 meters * √2
5. Calculate: Hypotenuse ≈ 11.31 meters

Therefore, the length of the diagonal path across the garden is approximately 11.31 meters.

Example 4: Solving for an Unknown Leg

Problem: One leg of a 45-45-90 triangle is 6 cm. Find the length of the other leg and the hypotenuse.

Solution:

1. Identify the known values: One leg is 6 cm.
2. Use the property of equal legs: The other leg is also 6 cm.
3. Apply the formula for the hypotenuse: Hypotenuse = Leg * √2
4. Substitute the values: Hypotenuse = 6 cm * √2
5. Calculate: Hypotenuse ≈ 8.49 cm

Therefore, the other leg is 6 cm, and the hypotenuse is approximately 8.49 cm.

Example 5: Using Trigonometry

Problem: Find the length of the hypotenuse of a 45-45-90 triangle if one leg is 12 units long.

Solution:

We can use trigonometric functions, specifically the sine or cosine function, since we know one leg and an angle.

1. Choose a trigonometric function: We can use sin(45°) = opposite/hypotenuse, where the opposite side is the leg.
2. Substitute the values: sin(45°) = 12/hypotenuse
3. Solve for the hypotenuse: hypotenuse = 12/sin(45°)
4. Since sin(45°) = 1/√2: hypotenuse = 12/(1/√2) = 12√2
5. Calculate: hypotenuse ≈ 16.97 units.

Example 6: A More Complex Problem

Problem: A right-angled triangle has angles of 45°, 45°, and 90°. The area of the triangle is 50 square units. Find the lengths of all three sides.

Solution:

1. Area Formula: The area of a triangle is (1/2) * base * height. In a 45-45-90 triangle, the base and height are equal (both legs).
2. Set up the equation: (1/2) * x * x = 50 (where x is the length of a leg)
3. Solve for x: x² = 100; x = 10 units
4. Find the hypotenuse: Hypotenuse = 10√2 ≈ 14.14 units

Therefore, the lengths of the sides are 10 units, 10 units, and approximately 14.14 units.

Advanced Applications and Problem Solving Strategies

Understanding 45-45-90 triangles opens doors to solving more complex geometric problems. Here are some strategies and advanced applications:

• Breaking down complex shapes: Often, larger geometric shapes can be broken down into simpler shapes, including 45-45-90 triangles. This allows you to apply the properties of the special triangle to solve for unknown lengths or angles within the larger shape.
• Using the Pythagorean Theorem: While the 45-45-90 triangle's properties provide a shortcut, the Pythagorean Theorem (a² + b² = c²) always applies to right-angled triangles, offering an alternative approach.
• Coordinate Geometry: 45-45-90 triangles frequently appear in coordinate geometry problems, particularly when dealing with lines with slopes of 1 or -1.
• Trigonometric Ratios: Using trigonometric functions like sine, cosine, and tangent can be a powerful tool, especially when dealing with angles and sides in a more general context than just the simple relationships within the 45-45-90 triangle.

Conclusion

Mastering the properties and applications of 45-45-90 triangles is essential for success in geometry and related fields. By understanding the relationship between its sides and angles and practicing with various examples, you'll build a strong foundation for tackling more advanced geometric problems and real-world applications. Remember to always visualize the problem, identify the known values, and choose the most efficient method to solve for the unknown variables. Consistent practice will significantly improve your ability to confidently work with these special right triangles.