Special Right Triangles Worksheet Answers

Mastering Special Right Triangles: A Comprehensive Guide with Worksheet Answers

Understanding special right triangles is crucial for success in geometry and trigonometry. These triangles, characterized by specific angle measures (30-60-90 and 45-45-90), possess unique side length ratios that simplify calculations and problem-solving significantly. This comprehensive guide provides a detailed explanation of these triangles, along with solved examples and answers to a practice worksheet, helping you master this essential concept.

Introduction: Unlocking the Secrets of Special Right Triangles

Special right triangles are a cornerstone of geometry. They are triangles with specific angle measurements that lead to predictable relationships between their side lengths. Knowing these relationships allows you to quickly determine unknown side lengths without relying on complex trigonometric functions like sine, cosine, and tangent. This significantly speeds up problem-solving in geometry and related fields. This article will focus on the two most common special right triangles: the 30-60-90 triangle and the 45-45-90 triangle. We'll explore their properties, provide detailed explanations, and offer a comprehensive worksheet with solutions to solidify your understanding.

Understanding the 30-60-90 Triangle

A 30-60-90 triangle is a right-angled triangle where the angles measure 30°, 60°, and 90°. The sides of this triangle follow a specific ratio:

• Shortest Side (opposite the 30° angle): This side is represented as 'x'.
• Longer Leg (opposite the 60° angle): This side is always √3 times the length of the shortest side, or x√3.
• Hypotenuse (opposite the 90° angle): The hypotenuse is twice the length of the shortest side, or 2x.

Therefore, the side ratio for a 30-60-90 triangle is 1 : √3 : 2.

Example 1: 30-60-90 Triangle Calculation

Let's say the shortest side (opposite the 30° angle) of a 30-60-90 triangle is 5 cm. We can find the lengths of the other two sides using the ratio:

• Shortest side (x): 5 cm
• Longer leg (x√3): 5√3 cm
• Hypotenuse (2x): 10 cm

Example 2: Finding the Shortest Side

If the hypotenuse of a 30-60-90 triangle is 12 cm, we can find the shortest side:

• Hypotenuse (2x): 12 cm
• Shortest side (x): 12 cm / 2 = 6 cm
• Longer leg (x√3): 6√3 cm

Understanding the 45-45-90 Triangle (Isosceles Right Triangle)

A 45-45-90 triangle is an isosceles right-angled triangle, meaning it has two equal angles (45° each) and one right angle (90°). The sides of this triangle also follow a specific ratio:

• Legs (opposite the 45° angles): Both legs are equal in length and are represented as 'x'.
• Hypotenuse (opposite the 90° angle): The hypotenuse is √2 times the length of either leg, or x√2.

Therefore, the side ratio for a 45-45-90 triangle is 1 : 1 : √2.

Example 3: 45-45-90 Triangle Calculation

If one leg of a 45-45-90 triangle is 8 cm, we can find the lengths of the other sides:

• Leg (x): 8 cm
• Other Leg (x): 8 cm
• Hypotenuse (x√2): 8√2 cm

Example 4: Finding a Leg Length

If the hypotenuse of a 45-45-90 triangle is 10√2 cm, we can find the length of each leg:

• Hypotenuse (x√2): 10√2 cm
• Leg (x): 10√2 cm / √2 = 10 cm
• Other Leg (x): 10 cm

Special Right Triangles Worksheet: Putting Your Knowledge to the Test

Here's a worksheet with problems to practice applying your knowledge of 30-60-90 and 45-45-90 triangles. Remember to use the ratios we've discussed.

Worksheet Questions:

Section 1: 30-60-90 Triangles

1. In a 30-60-90 triangle, the shortest side is 7 cm. Find the lengths of the other two sides.
2. The hypotenuse of a 30-60-90 triangle is 14 cm. Find the lengths of the two legs.
3. The longer leg of a 30-60-90 triangle is 6√3 cm. Find the lengths of the other two sides.
4. A right-angled triangle has angles 30°, 60°, and 90°. If the side opposite the 60° angle is 10 cm, what is the length of the hypotenuse?

Section 2: 45-45-90 Triangles

5. In a 45-45-90 triangle, one leg is 9 cm. Find the lengths of the other side.
6. The hypotenuse of a 45-45-90 triangle is 12√2 cm. Find the length of each leg.
7. A square has a diagonal of length 16 cm. What is the length of each side? (Hint: The diagonal of a square forms two 45-45-90 triangles)
8. A right-angled triangle has angles of 45°, 45°, and 90°. If the hypotenuse is 8 cm, what is the length of one leg?

Worksheet Answers:

Section 1: 30-60-90 Triangles

1. Shortest side: 7 cm, Longer leg: 7√3 cm, Hypotenuse: 14 cm
2. Shortest side: 7 cm, Longer leg: 7√3 cm
3. Shortest side: 6 cm, Hypotenuse: 12 cm
4. Hypotenuse: 20 cm

Section 2: 45-45-90 Triangles

5. Leg: 9 cm, Hypotenuse: 9√2 cm
6. Leg: 12 cm, Other Leg: 12 cm
7. Side: 8√2 cm
8. Leg: 4√2 cm

Further Applications and Advanced Concepts

Understanding special right triangles extends beyond basic calculations. They are fundamental in:

• Trigonometry: Special right triangles provide a simple way to understand the values of trigonometric functions for specific angles (30°, 45°, 60°).
• Geometry Proofs: These triangles are frequently used in geometric proofs, particularly those involving similar triangles and area calculations.
• Coordinate Geometry: The properties of special right triangles are useful in solving problems related to distance, slope, and equations of lines.
• Calculus: Understanding these triangles helps in visualizing and solving problems involving slopes and derivatives.

Conclusion: Mastering a Foundational Geometric Concept

Mastering special right triangles is a crucial step in building a strong foundation in geometry and trigonometry. By understanding the ratios of their sides and practicing problem-solving, you'll gain efficiency and confidence in tackling more complex geometric challenges. Remember, consistent practice is key. Use the worksheet and examples provided to hone your skills, and you'll soon find these triangles to be easily manageable tools in your mathematical arsenal. The more you practice, the more intuitive these relationships will become, and the more easily you'll be able to apply them to various problems. Remember to break down complex problems into smaller, more manageable parts, focusing on identifying the special right triangle within the larger geometric context. With consistent effort and practice, you will master this essential geometric concept.