Solving Triangle Sides: A Step-by-Step Guide

by TextBrain Team 45 views

Hey there, geometry enthusiasts! Today, we're diving into a classic triangle problem. We've got a triangle with two known angles (30° and 105°) and a height. Our mission? To find the length of the smallest side. Sounds like fun, right? Let's break it down, step by step, so you can ace similar problems in the future! We'll focus on understanding the concepts and applying them effectively.

Understanding the Problem: Triangle Angles and Heights

First things first, let's get a solid grip on what the problem throws at us. We're dealing with a triangle. Specifically, we know two of its interior angles: one is 30 degrees, and the other is a whopping 105 degrees. Knowing these angles is super helpful because we can easily figure out the third angle, which is always a good starting point in geometry problems. Additionally, we know the height drawn from the vertex of the largest angle, which is 105 degrees, and this height has a length of 4sqrt2\\sqrt{2}.

Now, the main question is: How do we find the smallest side? Remember that the smallest side of a triangle always faces the smallest angle. So, the side opposite the 30-degree angle is what we're hunting for. The height is drawn to one of the other two sides. The key here is to use trigonometry and the properties of special right triangles that are often formed when a height is drawn. Understanding the connection between angles, sides, and heights is crucial. It’s about spotting the relationships and using the right tools (trigonometry, in this case) to uncover the unknown.

Let's sketch out a rough drawing of the triangle, labeling the angles (A = 30°, B = 105°, C = 45°). Let's denote the height from vertex B as 'h', and the points where the height touches the opposite side as D. Now, you'll have two smaller right triangles: one with angles 30°, 60°, and 90° and the other with angles 45°, 45°, and 90°. The given height acts as one of the sides of the right angle for both these smaller triangles.

In essence, we are navigating the world of triangles, employing a blend of geometry, trigonometry, and a dash of clever problem-solving. Ready to proceed?

Step-by-Step Solution: Unveiling the Smallest Side

Alright, buckle up, guys! Here's how we're going to crack this problem. We'll break it down into manageable steps, making sure that the method is clear as a bell.

Step 1: Finding the Third Angle

Since the sum of angles in any triangle equals 180 degrees, we can calculate the third angle (let's call it angle C) using a simple subtraction: 180° - 105° - 30° = 45°. Now we know all three angles: 30°, 105°, and 45°. Knowing the third angle is useful for cross-checking, and also when applying the law of sines (although we won't need it directly in this solution).

Step 2: Drawing the Height and Forming Right Triangles

Let's draw the height from the vertex of the 105-degree angle to the opposite side. This height (let's call it h) divides our original triangle into two right-angled triangles. This is where things start to get interesting. With the help of this height, we can apply trigonometric ratios, such as sine, cosine, or tangent, to solve the problem.

Step 3: Focus on the Right Triangles

We have two right triangles. Let's focus on the one where we have a 30-degree angle. Knowing the height (4sqrt2\\sqrt{2}) and an angle (30°), we can use trigonometry to figure out the length of the adjacent side. In the right triangle, we have angle A as 30 degrees, and the height is the side opposite the angle. If we call the side we want to find 'x', we can use the sine function: sin(30°) = opposite/hypotenuse, so sin(30°) = h/AB, where AB is the hypotenuse. From this, you can solve for the sides of both right triangles.

Step 4: Applying Trigonometry

Now we can use trigonometry. In our first right triangle, we know one angle (30°) and the height (4sqrt2\\sqrt{2}). We want to find the length of the hypotenuse, which is one of the sides of our original triangle. Use the sine function, the sine of 30 degrees is 0.5, so we can set up the equation sin(30°) = opposite / hypotenuse, that is, 0.5 = 4sqrt2\\sqrt{2} / hypotenuse. By solving this equation, we find that the hypotenuse is equal to 8sqrt2\\sqrt{2}. Therefore, the longer side of the triangle is 8sqrt2\\sqrt{2}. Now, for the smaller triangle (the one with the 45-degree angle), we know the height is 4sqrt2\\sqrt{2}. Since it's a 45-45-90 triangle, the two legs are equal. Therefore, the other leg is also 4sqrt2\\sqrt{2}. Thus, the smallest side (opposite the 30-degree angle) is equal to 4sqrt2\\sqrt{2}.

Step 5: Identifying the Smallest Side

After we’ve done all the calculations, we identify the smallest side, which is the side opposite the 30-degree angle. Using the sine function, we can find the length of the side opposite the 30-degree angle. The side we seek is 4sqrt2\\sqrt{2}.

Detailed Calculations and Formulas

Let's dig into the formulas and calculations to solidify our understanding.

  • Finding the Third Angle: Always start by ensuring you know all three angles. This gives you a complete picture.

    • Formula: C = 180° - A - B.

  • Using Trigonometry: Trigonometry is the core here. We're dealing with right triangles, so the basic trigonometric functions come into play (sine, cosine, tangent).

    • Sine (sin): sin(angle) = Opposite / Hypotenuse
    • Cosine (cos): cos(angle) = Adjacent / Hypotenuse
    • Tangent (tan): tan(angle) = Opposite / Adjacent

  • Applying Trigonometric Ratios: These ratios help us connect angles with sides. When you know an angle and a side, you can find another side.

    • Example: sin(30°) = Opposite / Hypotenuse, therefore, Opposite = Hypotenuse * sin(30°).

  • 45-45-90 Triangles: A special type of right triangle, where the sides are in a ratio of 1:1:sqrt2\\sqrt{2}. If you know one leg, you know the other.

Let's write out the calculations for clarity:

  1. Third Angle: C = 180° - 105° - 30° = 45°
  2. Height = 4sqrt2\\sqrt{2}
  3. Right triangle with 30°: Let the side opposite the 30-degree angle be 'x'. Since the height forms a 45-45-90 triangle, we know that the side adjacent to the 45-degree angle is also 4sqrt2\\sqrt{2}.
  4. Finding the side opposite 30° (x): x = 4sqrt2\\sqrt{2}

Tips and Tricks for Triangle Problems

Alright, here are some pro tips to help you crush triangle problems consistently!

  • Always Draw a Diagram: A well-labeled diagram can make the problem much clearer.
  • Label Everything: Mark all angles, sides, and given information on your diagram. Use different colors to make things more visible.
  • Identify the Right Triangle: Look for right triangles. They open doors to trigonometry.
  • Special Triangles: Be aware of special triangles like 30-60-90 and 45-45-90. Their side ratios can save you time.
  • Double-Check Your Work: Ensure that your final answer makes sense in the context of the problem.

Remember, the key is practice. The more you work through these types of problems, the more natural the process will become. Don't be afraid to make mistakes – that's how we learn!

Conclusion: Mastering Triangle Side Calculations

So, there you have it! We successfully found the smallest side of our triangle by using our knowledge of angles, heights, and trigonometry. We've seen how to break down a complex problem into manageable parts, and now you're well-equipped to tackle similar problems. Remember to practice the concepts and try out various examples. Geometry problems become easier with each step.

Keep practicing, and you'll become a triangle-solving pro in no time! You've got this, guys!