45 45 90 Special Right Triangle Worksheet Answers

Decoding the 45-45-90 Special Right Triangle: A Comprehensive Guide with Worksheet Answers

The 45-45-90 triangle, also known as an isosceles right triangle, is a fundamental concept in geometry and trigonometry. Understanding its unique properties is crucial for solving various mathematical problems and lays the groundwork for more advanced concepts. This comprehensive guide will delve into the characteristics of a 45-45-90 triangle, provide a step-by-step approach to solving related problems, and offer solutions to a sample worksheet.

Understanding the 45-45-90 Triangle

A 45-45-90 triangle is a right-angled triangle where two of its angles measure 45 degrees each, and the third angle, naturally, measures 90 degrees. The key to solving problems involving these triangles lies in their unique side ratio. Due to its isosceles nature (two sides are equal), the legs (the sides adjacent to the right angle) are congruent. This inherent symmetry simplifies calculations considerably.

The Side Ratio: 1:1:√2

The crucial characteristic of a 45-45-90 triangle is its consistent side ratio. If we denote the length of the legs as 'x', then the length of the hypotenuse (the side opposite the right angle) will always be x√2. This ratio, 1:1:√2, is the cornerstone for solving problems involving 45-45-90 triangles. Remember this ratio; it's your key to unlocking many geometrical puzzles!

Solving Problems with 45-45-90 Triangles

Let's explore how to utilize the 1:1:√2 ratio to solve various problems. We'll tackle different scenarios to illustrate the versatility of this special triangle.

Scenario 1: Finding the Hypotenuse

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

Solution:

Since the legs are equal and have a length of 5 cm, we can directly apply the ratio:

• Leg 1 = 5 cm
• Leg 2 = 5 cm
• Hypotenuse = Leg * √2 = 5√2 cm

Therefore, the hypotenuse is 5√2 cm long. This is a simple application of the fundamental ratio.

Scenario 2: Finding the Legs

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

Solution:

This time, we know the hypotenuse and need to find the legs. We rearrange the ratio:

• Hypotenuse = Leg * √2
• Leg = Hypotenuse / √2

Substituting the given value:

• Leg = 10 cm / √2

To rationalize the denominator (remove the square root from the bottom), we multiply both the numerator and denominator by √2:

• Leg = (10√2) / (√2 * √2) = (10√2) / 2 = 5√2 cm

Therefore, each leg is 5√2 cm long.

Scenario 3: Applying Pythagorean Theorem

While the 1:1:√2 ratio is efficient, it's always beneficial to understand the connection to the Pythagorean theorem (a² + b² = c²). Let's verify our findings using this theorem.

Problem: Using the Pythagorean Theorem, verify the length of the hypotenuse in a 45-45-90 triangle with legs of 5 cm each.

Solution:

• a = 5 cm
• b = 5 cm
• c² = a² + b² = 5² + 5² = 25 + 25 = 50
• c = √50 = √(25 * 2) = 5√2 cm

The Pythagorean theorem confirms our earlier result: the hypotenuse is indeed 5√2 cm.

Advanced Applications and Problem Solving Strategies

The 45-45-90 triangle isn't limited to simple calculations. It forms the basis for solving complex geometrical problems involving:

• Trigonometric Functions: Understanding the relationship between angles and sides is vital for applying trigonometric functions (sine, cosine, tangent). In a 45-45-90 triangle, sin(45°) = cos(45°) = 1/√2 = √2/2.

• Coordinate Geometry: These triangles frequently appear in coordinate geometry problems, particularly when dealing with rotations and reflections.

• Area Calculation: The area of a 45-45-90 triangle is simply (1/2) * leg * leg = (1/2) * leg².

• 3D Geometry: Understanding the 45-45-90 triangle extends to 3D shapes like cubes and pyramids, facilitating the calculation of their dimensions and volumes.

Sample Worksheet and Solutions

Let's now work through a sample worksheet to reinforce the concepts discussed.

Worksheet:

1. A 45-45-90 triangle has legs of length 8 cm each. Find the length of the hypotenuse.
2. A 45-45-90 triangle has a hypotenuse of length 12√2 cm. Find the length of each leg.
3. Find the area of a 45-45-90 triangle with legs of length 6 cm each.
4. A square has a diagonal of length 14 cm. Find the length of each side. (Hint: The diagonal of a square forms two 45-45-90 triangles).
5. A right-angled isosceles triangle has an area of 50 cm². Calculate the length of its hypotenuse.

Solutions:

1. Hypotenuse = 8√2 cm
2. Leg = 12 cm
3. Area = 18 cm²
4. Side = 7√2 cm
5. Hypotenuse = 10√2 cm

Conclusion: Mastering the 45-45-90 Triangle

The 45-45-90 triangle, with its elegant simplicity and consistent ratio, is a fundamental building block in geometry and trigonometry. Mastering its properties and application will significantly enhance your problem-solving skills in various mathematical contexts. Remember the 1:1:√2 ratio, practice consistently, and you'll confidently tackle any challenge involving this special right triangle. By understanding its underlying principles and applying the methods outlined above, you’ll be well-equipped to solve a wide range of geometric problems. Keep practicing, and you'll develop a deep intuition for working with 45-45-90 triangles!