Triangle Side Lengths: Solving With Angles

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Hey math enthusiasts! Today, we're diving into a geometry problem that's all about triangles, angles, and sides. The question is: "A triangle has two angles measuring 31∘31^{\circ} and 68∘68^{\circ}. The side across from the third angle is 45 units long. What are the lengths of the other two sides?" Let's break it down and find out how to solve this, shall we?

Understanding the Problem: Triangle Basics

Alright, guys, before we jump into the calculations, let's make sure we're all on the same page about triangles. A triangle is a polygon with three sides and three angles. The sum of all interior angles in any triangle always equals 180∘180^{\circ}. This is super important because it helps us find missing angles! The problem gives us two angles, 31∘31^{\circ} and 68∘68^{\circ}, and the length of one side. We need to find the lengths of the other two sides. This is where the Law of Sines comes in handy. You can use this law to solve triangles when you know either two angles and a side (like in our problem) or two sides and an angle opposite one of those sides. Basically, the Law of Sines establishes a relationship between the angles of a triangle and the lengths of the sides opposite those angles. We're going to use this law to find the missing side lengths. Remember that it's all about matching angles with their opposite sides, and once we know the angle, we can find out the length of the opposite sides. If you are ever stuck on a math question, the best way is to start with the basics. That is what we are going to do today!

To solve this, we'll use the Law of Sines, which states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. In simpler terms, it means:

asin⁡(A)=bsin⁡(B)=csin⁡(C)\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}

Where:

  • a, b, and c are the side lengths of the triangle.
  • A, B, and C are the angles opposite sides a, b, and c, respectively.

Let's get started with this triangle by figuring out the third angle. Since we know two angles, we can easily find the third one. Then, we will use the Law of Sines to calculate the lengths of the other two sides. Easy, right?

Finding the Third Angle

First things first, we need to find the measure of the third angle. We know that the sum of the angles in a triangle is 180∘180^{\circ}. So, to find the missing angle, let's subtract the given angles from 180∘180^{\circ}. That is 180∘−31∘−68∘=81∘180^{\circ} - 31^{\circ} - 68^{\circ} = 81^{\circ}.

So, the third angle is 81∘81^{\circ}. Now that we have all three angles, we can move on to finding the lengths of the other two sides. Since we have all three angles and one side length, the Law of Sines is our best friend here.

Applying the Law of Sines

Now, let's use the Law of Sines to find the lengths of the other two sides. We know one side length (45 units) and its opposite angle (81∘81^{\circ}). We also know the other two angles (31∘31^{\circ} and 68∘68^{\circ}). Let's denote the sides as a, b, and c, where c = 45 units and the angle opposite c is 81∘81^{\circ}.

  1. Finding Side a (opposite the 31∘31^{\circ} angle):

    We can set up the Law of Sines as follows: asin⁡(31∘)=csin⁡(81∘)\frac{a}{\sin(31^{\circ})} = \frac{c}{\sin(81^{\circ})}

    Now, plug in the known values: asin⁡(31∘)=45sin⁡(81∘)\frac{a}{\sin(31^{\circ})} = \frac{45}{\sin(81^{\circ})}

    Solve for a: a=45⋅sin⁡(31∘)sin⁡(81∘)a = \frac{45 \cdot \sin(31^{\circ})}{\sin(81^{\circ})}

    Using a calculator: a≈45⋅0.51500.9877a \approx \frac{45 \cdot 0.5150}{0.9877}

    a≈23.5a \approx 23.5

  2. Finding Side b (opposite the 68∘68^{\circ} angle):

    Using the Law of Sines again: bsin⁡(68∘)=csin⁡(81∘)\frac{b}{\sin(68^{\circ})} = \frac{c}{\sin(81^{\circ})}

    Plug in the known values: bsin⁡(68∘)=45sin⁡(81∘)\frac{b}{\sin(68^{\circ})} = \frac{45}{\sin(81^{\circ})}

    Solve for b: b=45⋅sin⁡(68∘)sin⁡(81∘)b = \frac{45 \cdot \sin(68^{\circ})}{\sin(81^{\circ})}

    Using a calculator: b≈45⋅0.92720.9877b \approx \frac{45 \cdot 0.9272}{0.9877}

    b≈42.2b \approx 42.2

So, the lengths of the other two sides are approximately 23.5 units and 42.2 units, respectively. You can use the values to find out the other sides of the triangle. The Law of Sines is your best friend in this case, and it will take you far.

Conclusion: The Final Answer

Alright, folks, we've solved the problem! Using the Law of Sines, we found that the other two sides of the triangle have approximate lengths of 23.5 units and 42.2 units. So, in the multiple-choice options, the closest answer choices are A. 42.2 and B. 23.5. Therefore, the sides of the triangle are approximately 23.5 and 42.2.

This problem perfectly illustrates how the Law of Sines can be used to find missing side lengths in a triangle when you know the angles and one side. Keep practicing, and you'll become a pro at these types of problems in no time! Remember, the key is to understand the relationships between angles and sides and to apply the correct trigonometric formulas.

If you ever get stuck, go back to basics, and practice more problems! Keep the formulas in your brain, and you'll be good to go. Hopefully, this helps you with your math journey. Keep learning, keep practicing, and never be afraid to ask for help. Happy calculating!

Key Takeaways

  • The Law of Sines is crucial for solving triangles when you know angles and a side.
  • Always remember that the sum of angles in a triangle is 180∘180^{\circ}.
  • Make sure you align each side length to the correct angles.
  • Practice makes perfect! The more problems you solve, the better you'll get.

Now you know how to solve for triangle side lengths! Keep up the good work!