30-60-90 Triangles Worksheet Answer Key Explained
Table of Contents :
The 30-60-90 triangle is a special type of right triangle that has distinctive properties and ratios among its sides. Understanding these properties is fundamental in geometry and trigonometry, making it a crucial topic in mathematics education. This article will explore the characteristics of 30-60-90 triangles, provide insights into solving problems related to them, and explain the answer key for a typical worksheet on this topic. Let's dive into the world of 30-60-90 triangles! 📐
What is a 30-60-90 Triangle?
A 30-60-90 triangle is a right triangle where the angles measure 30 degrees, 60 degrees, and 90 degrees. The unique feature of this triangle is that it maintains consistent ratios between its side lengths. If you know the length of one side, you can easily determine the lengths of the other two sides using these ratios.
Side Length Ratios
In a 30-60-90 triangle, the lengths of the sides are in a fixed ratio of:
- 1 : √3 : 2
Where:
- The side opposite the 30-degree angle is the shortest side (1x),
- The side opposite the 60-degree angle is the medium length side (√3x),
- The side opposite the 90-degree angle (the hypotenuse) is the longest side (2x).
Example of Side Lengths
If we denote the shortest side (opposite the 30-degree angle) as 'x', the sides can be represented as:
| Angle | Opposite Side Length |
|---|---|
| 30 degrees | x |
| 60 degrees | x√3 |
| 90 degrees | 2x |
Solving 30-60-90 Triangles
When working on problems involving 30-60-90 triangles, you typically need to determine the lengths of the sides given one length. Below is a step-by-step approach to solve problems related to these triangles.
Step 1: Identify the Known Side
Before solving, determine which side is known. It could be the side opposite the 30°, 60°, or the hypotenuse. This knowledge will guide your calculations.
Step 2: Use the Ratios
Once you identify the known side, you can use the ratios provided above to find the unknown lengths.
Step 3: Calculate the Unknown Sides
For example, if you know that the side opposite the 30-degree angle is 5 units (x = 5):
-
The side opposite the 60-degree angle would be:
( 5\sqrt{3} \approx 8.66 ) units -
The hypotenuse would be:
( 2 \times 5 = 10 ) units
Important Note
When using the √3 value in calculations, it's often helpful to keep it in its radical form for exactness unless otherwise specified.
30-60-90 Triangle Worksheet Example
Let’s take a look at a worksheet that might focus on solving problems related to 30-60-90 triangles, and discuss the answer key.
Sample Problems
- Given that the shortest side (opposite 30 degrees) is 4 units, find the lengths of the other two sides.
- If the hypotenuse is 12 units, calculate the lengths of the remaining sides.
- Determine the length of the side opposite the 60-degree angle if it is known to be 6√3 units.
Answer Key Explained
| Problem Number | Known Side | Side Opposite 30° | Side Opposite 60° | Hypotenuse |
|---|---|---|---|---|
| 1 | 4 units | 4 units | 4√3 ≈ 6.93 units | 8 units |
| 2 | 12 units | 6 units | 6√3 ≈ 10.39 units | 12 units |
| 3 | 6√3 units | 2√3 ≈ 3.46 units | 6 units | 12 units |
Breakdown of the Solutions
-
Problem 1: The side opposite 30° is given as 4, so:
- Opposite 60°: ( 4\sqrt{3} \approx 6.93 )
- Hypotenuse: ( 2 \times 4 = 8 )
-
Problem 2: Given the hypotenuse:
- Opposite 30°: ( \frac{12}{2} = 6 )
- Opposite 60°: ( 6\sqrt{3} \approx 10.39 )
-
Problem 3: Knowing the side opposite the 60°:
- Opposite 30°: ( \frac{6\sqrt{3}}{\sqrt{3}} = 6 )
- Hypotenuse: ( 2 \times 6 = 12 )
Practical Applications of 30-60-90 Triangles
30-60-90 triangles are not just theoretical concepts; they have real-world applications in various fields. Here are a few examples:
- Architecture: Designers often use these triangles to ensure the stability and aesthetic proportion of structures.
- Engineering: Engineers apply the principles of 30-60-90 triangles when designing components that require precision.
- Art and Design: Artists and designers use these ratios to create visually appealing compositions.
Conclusion
Understanding 30-60-90 triangles is vital for students as it lays the foundation for more complex geometric concepts. The consistent ratios of sides allow for easy calculations, which can be applied in various mathematical and practical scenarios. Familiarizing yourself with these properties through worksheets and answer keys will enhance your proficiency in geometry and trigonometry, equipping you with skills applicable in academia and beyond. Happy studying! 📚✨