Solving for the Hypotenuse in a 30-60-90 Triangle
In geometry, 30-60-90 triangles are particularly interesting due to their unique properties and the consistent ratios of their side lengths. This article will guide you through solving for the hypotenuse of a 30-60-90 triangle when given the length of the longer leg, providing step-by-step calculations and insightful explanations.
The Properties of a 30-60-90 Triangle
In any 30-60-90 triangle, the sides are in a specific ratio. Let's denote the shorter leg (opposite the 30-degree angle) as x. The longer leg (opposite the 60-degree angle) is then x √3, and the hypotenuse (opposite the 90-degree angle) is 2x.
Solving for the Hypotenuse When the Longer Leg is Given
This article will explore a specific problem: if the longer leg of a 30-60-90 triangle is 13, what is the length of the hypotenuse?
Step-by-Step Solution
Step 1: Establish the RatioThe given information states that the longer leg is 13. This leg corresponds to the x √3 part of the 30-60-90 triangle ratio. Therefore, we set up the equation: x √3 13 Step 2: Solve for x
To isolate x, divide both sides of the equation by √3: x 13 / √3 Step 3: Simplify the Expression
Simplify the expression by multiplying both the numerator and the denominator by √3 to rationalize the denominator: x (13 √3) / 3 Step 4: Calculate the Hypotenuse
Since the hypotenuse is equal to 2x, substitute the value of x into this expression: Hypotenuse 2 * (13 √3) / 3
Simplify this expression:
Hypotenuse 26 √3 / 3 ≈ 15
Understanding the Ratios in a 30-60-90 Triangle
By drawing a 30-60-90 triangle and labeling the sides based on the given angles (30°, 60°, and 90°), you can visually confirm the consistent ratios. For instance, if the shorter leg is labeled as x, the longer leg will be x √3, and the hypotenuse will be 2x.
Practice and Memorization of Trigonometric Ratios
Mastering the ratios of 30-60-90 triangles and other special triangles (e.g., 45-45-90) is crucial for solving more complex trigonometric and geometric problems. Here, we recommend a few key steps:
Always draw a picture of the triangle.Ensure that the correct angles and sides are labeled accurately. Utilize your knowledge of sine and cosine values for these specific angles.
For example, sin 60° √3 / 2 and cos 60° 1 / 2. Practice similar problems to reinforce your understanding and problem-solving skills.
Conclusion
Understanding and solving 30-60-90 triangles can significantly enhance your problem-solving abilities in geometry and trigonometry. By following the steps outlined in this article and practicing similar problems, you can confidently solve for the hypotenuse or other side lengths given specific conditions.