The 30-60-90 triangle is a special right triangle with angles measuring 30, 60, and 90 degrees. This triangle is commonly used in trigonometry and geometry, and it’s essential to understand its properties to solve various math problems. In this worksheet answer key, we’ll explore the 30-60-90 triangle and provide answers to some common problems.
Understanding the 30-60-90 Triangle
The 30-60-90 triangle has some unique properties that make it useful for solving trigonometric problems. Here are some key things to remember:
- The side opposite the 30-degree angle is half the length of the hypotenuse.
- The side opposite the 60-degree angle is 3⁄2 times the length of the side opposite the 30-degree angle.
- The ratio of the side lengths is 1:√3:2, where 1 is the side opposite the 30-degree angle, √3 is the side opposite the 60-degree angle, and 2 is the hypotenuse.
Solving 30-60-90 Triangle Problems
Now, let’s solve some common problems involving the 30-60-90 triangle.
Problem 1: Finding the Length of a Side
In a 30-60-90 triangle, the hypotenuse has a length of 10 inches. What is the length of the side opposite the 30-degree angle?
Answer: Since the side opposite the 30-degree angle is half the length of the hypotenuse, the answer is 5 inches.
Problem 2: Finding the Length of a Side
In a 30-60-90 triangle, the side opposite the 60-degree angle has a length of 6 inches. What is the length of the hypotenuse?
Answer: Since the ratio of the side lengths is 1:√3:2, we can set up a proportion to find the length of the hypotenuse:
1:√3 = 6:x
Solving for x, we get:
x = 2 × 6 = 12 inches
Problem 3: Finding the Measure of an Angle
In a 30-60-90 triangle, the side opposite the 30-degree angle has a length of 4 inches, and the hypotenuse has a length of 8 inches. What is the measure of the other acute angle?
Answer: Since the triangle is a 30-60-90 triangle, the other acute angle must be 60 degrees.
More Practice Problems
Here are some more practice problems to help you master the 30-60-90 triangle:
- In a 30-60-90 triangle, the hypotenuse has a length of 15 inches. What is the length of the side opposite the 60-degree angle?
- In a 30-60-90 triangle, the side opposite the 30-degree angle has a length of 3 inches. What is the length of the hypotenuse?
- In a 30-60-90 triangle, the side opposite the 60-degree angle has a length of 9 inches. What is the length of the side opposite the 30-degree angle?
Answers:
- 7.5 inches
- 6 inches
- 3 inches
Notes:
📝 Note: When solving 30-60-90 triangle problems, make sure to identify the ratio of the side lengths and use it to find the length of the unknown side.
Conclusion:
The 30-60-90 triangle is a fundamental concept in trigonometry and geometry. By understanding its properties and practicing with various problems, you’ll become more proficient in solving trigonometric problems. Remember to identify the ratio of the side lengths and use it to find the length of the unknown side.
FAQ Section:
What is a 30-60-90 triangle?
+A 30-60-90 triangle is a special right triangle with angles measuring 30, 60, and 90 degrees.
What is the ratio of the side lengths in a 30-60-90 triangle?
+The ratio of the side lengths is 1:√3:2, where 1 is the side opposite the 30-degree angle, √3 is the side opposite the 60-degree angle, and 2 is the hypotenuse.
How do I find the length of a side in a 30-60-90 triangle?
+Use the ratio of the side lengths to find the length of the unknown side.
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