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 type of right-angled triangle with unique properties that make it frequently encountered in various mathematical and real-world applications. Understanding its characteristics and how to solve problems involving it is crucial for anyone studying geometry or trigonometry. This comprehensive guide will explore numerous examples of 45-45-90 triangles, providing detailed solutions and explanations to solidify your understanding.

Understanding the 45-45-90 Triangle

A 45-45-90 triangle is characterized by its angles: two angles measuring 45 degrees each and one right angle (90 degrees). This specific angle configuration leads to an important relationship between its sides:

• Isosceles Nature: The two legs (sides opposite the 45-degree angles) are congruent (equal in length).
• Hypotenuse Relationship: The hypotenuse (the side opposite the 90-degree angle) is √2 times the length of each leg. This is derived from the Pythagorean theorem (a² + b² = c², where 'a' and 'b' are legs and 'c' is the hypotenuse). Since a = b in a 45-45-90 triangle, the relationship becomes a² + a² = c², simplifying to c = a√2.

This consistent relationship allows for quick calculations once one side is known. Let's delve into specific examples.

Example Problems and Solutions

Let's explore various scenarios involving 45-45-90 triangles, progressing in complexity. Each problem will provide a step-by-step solution to illustrate the application of the properties discussed.

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) are 5 cm.
2. Apply the hypotenuse relationship: Hypotenuse (c) = leg * √2
3. Calculate: c = 5 cm * √2 ≈ 7.07 cm

Answer: The length of the hypotenuse is approximately 7.07 cm.

Example 2: Finding the Legs

Problem: The hypotenuse of a 45-45-90 triangle measures 12 inches. Determine the length of each leg.

Solution:

1. Identify the known value: The hypotenuse (c) is 12 inches.
2. Apply the hypotenuse relationship (reversed): Leg (a) = hypotenuse / √2
3. Calculate: a = 12 inches / √2 ≈ 8.49 inches

Answer: Each leg measures approximately 8.49 inches.

Example 3: Area Calculation

Problem: Calculate the area of a 45-45-90 triangle with a hypotenuse of 10 meters.

Solution:

1. Find the leg length: Leg (a) = hypotenuse / √2 = 10 m / √2 ≈ 7.07 m
2. Calculate the area: Area = (1/2) * base * height. In a 45-45-90 triangle, the base and height are equal to the leg length.
3. Calculate: Area = (1/2) * 7.07 m * 7.07 m ≈ 25 square meters

Answer: The area of the triangle is approximately 25 square meters.

Example 4: Real-world Application - Diagonal of a Square

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

Solution:

1. Recognize the 45-45-90 triangle: The diagonal of a square divides it into two congruent 45-45-90 triangles. The legs of each triangle are the sides of the square.
2. Apply the hypotenuse relationship: Diagonal = side * √2
3. Calculate: Diagonal = 8 feet * √2 ≈ 11.31 feet

Answer: The length of the diagonal across the garden is approximately 11.31 feet.

Example 5: More Complex Problem Involving Other Triangles

Problem: A larger right-angled triangle has a hypotenuse of 20 cm. One of its angles is 45 degrees. Find the lengths of the other two sides.

Solution:

1. Identify the triangle type: Since one angle is 45 degrees and the triangle is right-angled, it's a 45-45-90 triangle.
2. Apply the hypotenuse relationship (reversed): Leg (a) = hypotenuse / √2 = 20 cm / √2 ≈ 14.14 cm
3. The other leg is equal: Since it’s a 45-45-90 triangle, the other leg also measures approximately 14.14 cm.

Answer: The other two sides measure approximately 14.14 cm each.

Example 6: Trigonometric Functions

Problem: In a 45-45-90 triangle with a leg of length 'x', find the values of sin(45°), cos(45°), and tan(45°).

Solution:

1. Hypotenuse: The hypotenuse is x√2.
2. Sin(45°): Opposite/Hypotenuse = x / (x√2) = 1/√2 = √2/2
3. Cos(45°): Adjacent/Hypotenuse = x / (x√2) = 1/√2 = √2/2
4. Tan(45°): Opposite/Adjacent = x / x = 1

Answer: sin(45°) = cos(45°) = √2/2 and tan(45°) = 1.

Advanced Applications and Considerations

The 45-45-90 triangle's properties extend beyond simple calculations. They are fundamental to:

• Coordinate Geometry: Understanding distances and slopes of lines.
• Calculus: Derivations and integrations involving geometric shapes.
• Engineering and Architecture: Designing structures with precise angles and dimensions.
• Physics: Analyzing forces and vectors.

Practical Tips for Solving 45-45-90 Triangle Problems

• Memorize the ratio: The ratio of sides (leg:leg:hypotenuse) is 1:1:√2. This is the key to quick solutions.
• Draw a diagram: A visual representation helps clarify the problem and identify the known and unknown values.
• Use the Pythagorean theorem as a backup: If you get stuck, the Pythagorean theorem always provides a reliable method for solving right-angled triangles.
• Practice consistently: Solving various problems will strengthen your understanding and improve your problem-solving speed.

Conclusion

The 45-45-90 triangle, despite its simplicity, is a powerful geometric concept with wide-ranging applications. By understanding its unique properties and practicing problem-solving techniques, you can confidently tackle various mathematical and real-world challenges involving this special type of right-angled triangle. Remember to focus on understanding the underlying principles rather than rote memorization to truly master this important geometric tool.