Special Triangles Worksheet With Answers

Special Triangles Worksheet: A thorough look with Answers

This worksheet will explore the properties and applications of two crucial types of special triangles: 30-60-90 triangles and 45-45-90 triangles. Understanding these triangles is fundamental in trigonometry, geometry, and various engineering and architectural applications. This guide will provide detailed explanations, example problems, and solutions to help you master these important geometric concepts.

Introduction to Special Triangles

Special right triangles are characterized by their specific angle measurements and the resulting ratios between their side lengths. Also, these consistent ratios allow for quick calculations without relying on complex trigonometric functions like sine, cosine, and tangent, particularly useful for solving problems quickly and efficiently. Mastering these triangles is a cornerstone of understanding more advanced geometrical concepts.

30-60-90 Triangles: The Fundamentals

A 30-60-90 triangle is a right-angled triangle with angles measuring 30°, 60°, and 90°. The side lengths follow a specific ratio:

• Shortest side (opposite 30°): This is considered the 'x' value.
• Longer leg (opposite 60°): This side is always √3 times the length of the shortest side (x√3).
• Hypotenuse (opposite 90°): This is always twice the length of the shortest side (2x).

This ratio, x : x√3 : 2x, is crucial for solving problems involving 30-60-90 triangles. Remember, this ratio only holds true for triangles with these specific angles.

Example Problems: 30-60-90 Triangles

Problem 1: A 30-60-90 triangle has a hypotenuse of 10 cm. Find the lengths of the other two sides.

Solution:

1. Identify the known: The hypotenuse (2x) = 10 cm.
2. Solve for x: 2x = 10, therefore x = 5 cm (this is the shortest side).
3. Calculate the other side: The side opposite the 60° angle is x√3 = 5√3 cm.

Which means, the sides of the triangle are 5 cm, 5√3 cm, and 10 cm.

Problem 2: A 30-60-90 triangle has a side opposite the 60° angle measuring 8√3 cm. Find the lengths of the other two sides.

Solution:

1. Identify the known: The side opposite the 60° angle (x√3) = 8√3 cm.
2. Solve for x: x√3 = 8√3, therefore x = 8 cm (this is the shortest side).
3. Calculate the hypotenuse: The hypotenuse is 2x = 2 * 8 cm = 16 cm.

Which means, the sides of the triangle are 8 cm, 8√3 cm, and 16 cm.

45-45-90 Triangles: The Isosceles Right Triangle

A 45-45-90 triangle is an isosceles right-angled triangle. This means it has two equal angles (45°) and one right angle (90°). The side lengths follow a simple ratio:

• Legs (opposite 45° angles): Both legs are equal in length, represented by 'x'.
• Hypotenuse (opposite 90°): The hypotenuse is always √2 times the length of one leg (x√2).

The ratio is x : x : x√2. This ratio makes calculations for 45-45-90 triangles remarkably straightforward.

Example Problems: 45-45-90 Triangles

Problem 1: A 45-45-90 triangle has a leg measuring 7 cm. Find the length of the hypotenuse.

Solution:

1. Identify the known: One leg (x) = 7 cm.
2. Calculate the hypotenuse: The hypotenuse is x√2 = 7√2 cm.

Which means, the sides of the triangle are 7 cm, 7 cm, and 7√2 cm.

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Problem 2: A 45-45-90 triangle has a hypotenuse of 12√2 cm. Find the length of each leg.

Solution:

1. Identify the known: The hypotenuse (x√2) = 12√2 cm.
2. Solve for x: x√2 = 12√2, therefore x = 12 cm.

So, the legs of the triangle are 12 cm each.

Worksheet Exercises: 30-60-90 and 45-45-90 Triangles

Now, let's put your knowledge to the test with some practice problems. Remember to show your work for each problem.

Section 1: 30-60-90 Triangles

1. A 30-60-90 triangle has a shortest side of 6 inches. Find the lengths of the other two sides.
2. A 30-60-90 triangle has a hypotenuse of 18 cm. Find the lengths of the other two sides.
3. A 30-60-90 triangle has a side opposite the 60° angle of 10√3 feet. Find the lengths of the other two sides.
4. The longest side of a 30-60-90 triangle is 24 meters. What are the lengths of the two shorter sides?
5. In a 30-60-90 triangle, the side opposite the 30° angle is 5√3 units. What is the length of the hypotenuse?

Section 2: 45-45-90 Triangles

6. A 45-45-90 triangle has a leg of 9 mm. Find the length of the hypotenuse.
7. A 45-45-90 triangle has a hypotenuse of 14√2 yards. Find the length of each leg.
8. The legs of a 45-45-90 triangle are both 11 inches long. What is the length of the hypotenuse?
9. The hypotenuse of a 45-45-90 triangle measures 8 cm. Calculate the length of each leg.
10. A square has a diagonal of 10√2 feet. What is the length of each side of the square? (Hint: consider the triangles formed by the diagonal).

Answers to Worksheet Exercises

Section 1: 30-60-90 Triangles

1. Shortest side: 6 inches; Other side: 6√3 inches; Hypotenuse: 12 inches
2. Shortest side: 9 cm; Other side: 9√3 cm; Hypotenuse: 18 cm
3. Shortest side: 10 feet; Hypotenuse: 20 feet
4. Shortest side: 12 meters; Other side: 12√3 meters
5. Hypotenuse: 10√3 units

Section 2: 45-45-90 Triangles

6. Hypotenuse: 9√2 mm
7. Length of each leg: 14 yards
8. Hypotenuse: 11√2 inches
9. Length of each leg: 8/√2 = 4√2 cm
10. Length of each side: 10 feet

Further Exploration and Applications

Understanding special triangles is a fundamental skill that extends far beyond basic geometry. Their applications are widespread:

• Trigonometry: Special triangles provide a simplified approach to solving trigonometric problems, particularly before the use of calculators.
• Engineering and Architecture: These triangles are frequently used in structural design, surveying, and other fields requiring precise calculations.
• Computer Graphics and Game Development: The principles of special triangles are essential for creating realistic 2D and 3D graphics.

This worksheet and its solutions are designed to equip you with the necessary tools to confidently tackle problems involving 30-60-90 and 45-45-90 triangles. So remember to practice consistently to reinforce your understanding. Also, the more you work with these triangles, the more intuitive their properties will become. Good luck!