Special Right Triangles Worksheet Answer Key 30-60-90

Mastering the 30-60-90 Triangle: A Comprehensive Guide with Worksheet and Answer Key

Understanding special right triangles is crucial for success in geometry and trigonometry. Among these, the 30-60-90 triangle holds a special place due to its predictable side ratios, making calculations significantly easier. This comprehensive guide will delve into the properties of 30-60-90 triangles, provide practical examples, and offer a worksheet with an answer key to solidify your understanding. This article will cover everything from the fundamental theorem to advanced applications, ensuring you can confidently tackle any problem involving this special triangle.

Meta Description: Learn the properties of 30-60-90 triangles, master their ratios, and solve various problems with our detailed guide, including a practice worksheet and answer key. Perfect for geometry and trigonometry students!

Understanding the 30-60-90 Triangle Theorem

The 30-60-90 triangle theorem states that the sides of a 30-60-90 triangle are always in a specific ratio: 1 : √3 : 2. This ratio refers to the lengths of the sides opposite to the 30°, 60°, and 90° angles, respectively. Let's break this down:

• The shortest side (opposite the 30° angle): This side is always considered to have a length of '1' or 'x' (a variable representing the shortest side).

• The medium side (opposite the 60° angle): This side is always √3 times the length of the shortest side. So, if the shortest side is 'x', the medium side is 'x√3'.

• The hypotenuse (opposite the 90° angle): This side is always twice the length of the shortest side. Therefore, if the shortest side is 'x', the hypotenuse is '2x'.

This constant ratio simplifies calculations significantly. Instead of using trigonometric functions like sine, cosine, and tangent for every problem, you can directly apply this ratio to find the lengths of unknown sides.

Deriving the 30-60-90 Triangle Ratio

The 30-60-90 triangle ratio is derived from an equilateral triangle. Consider an equilateral triangle with all sides of length '2x'. By drawing an altitude from one vertex to the midpoint of the opposite side, we bisect the equilateral triangle into two congruent 30-60-90 triangles.

The altitude divides the base into two equal segments, each of length 'x'. Using the Pythagorean theorem (a² + b² = c²) on one of the 30-60-90 triangles, we can find the length of the altitude:

x² + h² = (2x)²

h² = 4x² - x²

h² = 3x²

h = x√3

This confirms the ratio: shortest side = x, medium side = x√3, and hypotenuse = 2x.

Practical Applications and Examples

Let's explore several examples to illustrate how to apply the 30-60-90 triangle theorem:

Example 1: Finding Missing Sides

A 30-60-90 triangle has a hypotenuse of length 10. Find the lengths of the other two sides.

• Solution: Since the hypotenuse is 2x = 10, then x = 5 (the shortest side).
• The side opposite the 60° angle is x√3 = 5√3.

Therefore, the sides are 5, 5√3, and 10.

Example 2: Finding the Hypotenuse

A 30-60-90 triangle has a side of length 4 opposite the 30° angle. Find the length of the hypotenuse.

• Solution: The shortest side is x = 4.
• The hypotenuse is 2x = 2 * 4 = 8.

Example 3: Finding the Side Opposite 60°

A 30-60-90 triangle has a hypotenuse of length 12. Find the length of the side opposite the 60° angle.

• Solution: The hypotenuse is 2x = 12, so x = 6.
• The side opposite the 60° angle is x√3 = 6√3.

Example 4: Real-World Application – Geometry Problem

Imagine a ramp inclined at 30° to the horizontal. If the ramp's horizontal length is 15 feet, how high is the ramp's vertical rise? This forms a 30-60-90 triangle where the horizontal length is the side opposite the 60° angle.

• Solution: The horizontal length (opposite the 60° angle) is x√3 = 15.
• Solving for x (shortest side): x = 15/√3 = 5√3.
• The vertical rise (shortest side) is x = 5√3 feet.

Advanced Applications: Area and Perimeter Calculations

The 30-60-90 triangle theorem also facilitates the calculation of the area and perimeter.

Area: The area of a triangle is given by (1/2) * base * height. In a 30-60-90 triangle, the base can be the side opposite the 60° angle (x√3), and the height is the shortest side (x). Therefore, the area is:

(1/2) * x√3 * x = (√3/2)x²

Perimeter: The perimeter is simply the sum of all three sides: x + x√3 + 2x = 3x + x√3 = x(3 + √3).

30-60-90 Triangle Worksheet

Here's a worksheet to test your understanding. Remember to apply the 1:√3:2 ratio consistently. Try to solve these problems before checking the answer key below.

Problems:

1. A 30-60-90 triangle has a shortest side of length 7. Find the lengths of the other two sides.
2. A 30-60-90 triangle has a hypotenuse of length 18. Find the lengths of the other two sides.
3. A 30-60-90 triangle has a side of length 6√3 opposite the 60° angle. Find the lengths of the other two sides.
4. Find the area of a 30-60-90 triangle with a hypotenuse of 14.
5. Find the perimeter of a 30-60-90 triangle with a shortest side of length 5.
6. A right triangle has angles of 30°, 60°, and 90°. The side opposite the 30° angle is 8 cm. Calculate the lengths of the other two sides.
7. A 30-60-90 triangle has an area of 27√3 square centimeters. What is the length of its hypotenuse?
8. A flagpole casts a shadow that is 20 meters long. The angle of elevation from the tip of the shadow to the top of the flagpole is 30°. How tall is the flagpole? (Hint: Consider this a 30-60-90 triangle).

30-60-90 Triangle Worksheet Answer Key

1. Shortest side: 7; Side opposite 60°: 7√3; Hypotenuse: 14
2. Shortest side: 9; Side opposite 60°: 9√3; Hypotenuse: 18
3. Shortest side: 6; Side opposite 30°: 6; Hypotenuse: 12
4. Area: 49√3/2
5. Perimeter: 15 + 5√3
6. Side opposite 60°: 8√3 cm; Hypotenuse: 16 cm
7. Hypotenuse: 12 cm
8. Flagpole height: 20/√3 meters or (20√3)/3 meters

Conclusion

The 30-60-90 triangle is a fundamental concept in geometry and trigonometry. Mastering its properties and ratios provides a powerful tool for solving a wide range of problems efficiently. By understanding the theorem, practicing with examples, and completing the provided worksheet, you'll build a strong foundation for tackling more complex geometric challenges. Remember to always visualize the triangle and carefully identify which side corresponds to each angle. Consistent practice is key to achieving fluency in solving 30-60-90 triangle problems.