45 45 90 Triangle Worksheet Answer Key

45-45-90 Triangle Worksheet Answer Key: A Comprehensive Guide

The 45-45-90 triangle, also known as an isosceles right triangle, is a fundamental concept in trigonometry and geometry. Understanding its properties is crucial for solving various mathematical problems. This comprehensive guide provides a detailed explanation of 45-45-90 triangles, including their characteristics, formulas, and a walkthrough of solving problems, effectively serving as a comprehensive answer key for common worksheet questions.

Understanding the 45-45-90 Triangle

A 45-45-90 triangle is a special right triangle characterized by its angles: two angles measure 45 degrees each, and one angle measures 90 degrees (a right angle). Because two of its angles are equal, it's also an isosceles triangle, meaning two of its sides are equal in length. This unique structure leads to predictable relationships between its sides, making calculations simpler than in general right triangles.

Key Characteristics:

• Two equal angles: 45° and 45°
• One right angle: 90°
• Two equal sides (legs): These sides are opposite the 45° angles.
• Hypotenuse: The side opposite the 90° angle. It's always longer than the legs.

The 45-45-90 Triangle Theorem

The relationship between the sides of a 45-45-90 triangle is defined by a specific theorem:

Theorem: In a 45-45-90 triangle, the length of the hypotenuse is √2 times the length of each leg.

This theorem is derived from the Pythagorean theorem (a² + b² = c²), where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse. Since the legs are equal in length (let's call them 'x'), the theorem simplifies to:

x² + x² = c² 2x² = c² c = x√2

This formula is the cornerstone for solving problems involving 45-45-90 triangles.

Solving Problems: A Step-by-Step Approach

Let's explore various scenarios and demonstrate how to apply the 45-45-90 triangle theorem to find missing side lengths.

Scenario 1: Given one leg, find the hypotenuse.

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

Solution:

1. Identify the known value: Leg length (x) = 5 cm
2. Apply the theorem: Hypotenuse (c) = x√2
3. Substitute and solve: c = 5√2 cm

Therefore, the hypotenuse is 5√2 cm. You can leave the answer in this form (exact value) or approximate it using a calculator (approximately 7.07 cm).

Scenario 2: Given the hypotenuse, find the legs.

Problem: A 45-45-90 triangle has a hypotenuse of length 8√2 inches. Find the length of each leg.

Solution:

1. Identify the known value: Hypotenuse (c) = 8√2 inches
2. Apply the theorem (rearrange): x = c/√2
3. Substitute and solve: x = (8√2)/√2 = 8 inches

Therefore, each leg has a length of 8 inches.

Scenario 3: Using Trigonometry

While the theorem provides a direct solution, trigonometric functions (sine, cosine, tangent) can also be applied. Since we know the angles, we can use these functions:

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

Remember that in a 45-45-90 triangle, the opposite and adjacent sides (legs) are equal. sin(45°) = cos(45°) = 1/√2 = √2/2. tan(45°) = 1.

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

Solution using sine:

1. Identify the known value: Hypotenuse = 10 meters
2. Use sine function: sin(45°) = opposite/hypotenuse
3. Substitute and solve: (√2/2) = opposite/10
4. Solve for the opposite side: opposite = 10(√2/2) = 5√2 meters

Scenario 4: More Complex Problems

Often, 45-45-90 triangles are embedded within larger geometric figures. Solving these problems requires a multi-step approach, combining the 45-45-90 theorem with other geometric principles.

Problem: A square has a diagonal of length 12 cm. Find the length of each side.

Solution:

1. Recognize the 45-45-90 triangles: The diagonal of a square divides it into two congruent 45-45-90 triangles.
2. Identify the known value: Hypotenuse = 12 cm
3. Apply the theorem (rearrange): x = c/√2
4. Substitute and solve: x = 12/√2 = 6√2 cm

Therefore, each side of the square is 6√2 cm long.

Common Worksheet Questions and Solutions

Here are examples of typical problems found in 45-45-90 triangle worksheets, along with detailed solutions:

Problem 1: Find the length of the hypotenuse of a 45-45-90 triangle with legs of length 7 units.

Solution: Hypotenuse = 7√2 units

Problem 2: A 45-45-90 triangle has a hypotenuse of length 14√2 meters. What is the length of each leg?

Solution: Each leg = 14 meters

Problem 3: An isosceles right triangle has an area of 50 square inches. Find the length of its legs.

Solution:

1. Area of a triangle: (1/2) * base * height
2. In this case: (1/2) * x * x = 50 (where x is the length of each leg)
3. Solve for x: x² = 100, x = 10 inches

Problem 4: Find the perimeter of a 45-45-90 triangle with a leg of length 6 cm.

Solution:

1. Hypotenuse: 6√2 cm
2. Perimeter: 6 + 6 + 6√2 = 12 + 6√2 cm

Problem 5: A square has an area of 64 square feet. What is the length of its diagonal?

Solution:

1. Side length of square: √64 = 8 feet
2. Diagonal: 8√2 feet

Problem 6: Two legs of a right-angled triangle are equal and measure 10 cm each. Find the length of its hypotenuse. Is this a 45-45-90 triangle?

Solution:

1. Apply Pythagorean Theorem: 10² + 10² = hypotenuse²
2. Hypotenuse: √200 = 10√2 cm
3. Yes, it's a 45-45-90 triangle because it has two equal legs and a right angle.

Advanced Applications and Real-World Examples

The 45-45-90 triangle is more than just a classroom concept; it has practical applications in various fields:

• Construction and Engineering: Calculating diagonal measurements, roof angles, and support structures.
• Architecture: Designing symmetrical structures and determining optimal dimensions.
• Computer Graphics and Game Development: Creating realistic 3D models and environments.
• Navigation and Surveying: Determining distances and angles in land surveying.

Understanding the properties and solving problems related to 45-45-90 triangles is essential for success in many areas of mathematics and its practical applications. Mastering this concept builds a strong foundation for more advanced geometric and trigonometric studies. By consistently practicing with various problems and utilizing the strategies outlined in this comprehensive guide, you'll develop proficiency in solving these types of problems effectively. Remember to always check your work and consider alternative methods to ensure accuracy and deepen your understanding.