How to Find the Missing Side of a Triangle: A Comprehensive Guide

Introduction

Triangle problems often cause frustration for many students and individuals who need to perform calculations involving triangles. A common problem for many is finding the missing side of a triangle. In this article, we will explore the different methods of finding the missing side of a triangle, from the basic Pythagorean Theorem to the advanced Laws of Cosines and Sines. We will also provide numerous examples and practice problems so that you can perfect your techniques. Knowing how to do this is important because the missing side of a triangle is a critical component of triangle puzzle solutions, engineering, and construction.

Unraveling the Mystery: A Guide to Finding the Missing Side of a Triangle

The Pythagorean Theorem is the most basic method for finding the missing side of a triangle. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is the sum of the squares of the other two sides. By applying this theorem one can easily find one of the missing sides of a right triangle if the other two sides are known. Here is an example:

Example: Find the value of ‘x’ for the following right triangle if one side is 7 meters and the other side is 24 meters.

Applying the Pythagorean Theorem: 7² + x² = 24²

49 + x² = 576

x = 23 meters (rounded to nearest whole number)

Practice this method with some more examples:

Practice Problem 1: Find the missing side ‘a’ for the following right triangle if one of the other sides is 5 and the hypotenuse is 13.

Practice Problem 2: Find the missing side ‘b’ for the following right triangle if one of the other sides is 15 and the hypotenuse is 17.

Math Made Easy: How to Find the Missing Side of Any Triangle

The Law of Cosines and the Law of Sines are used for any kind of triangle, including right triangles. Essentially, these laws use ratios of sides and angles in order to find the missing side. The Law of Cosines involves the relationship between the sides and angles, and the Law of Sines involves the relationship between the angles and sides. Here is an example of how to apply the Law of Cosines to find the missing side of a triangle:

Example: Find the length of missing side ‘c’ for the triangle below.

Applying the Law of Cosines: c² = a² + b² – 2abcos(C)

c² = 6² + 8² – 2(6)(8)cos(40°) (using cosine rule)

c = 6.9 (rounded to nearest tenth of a unit)

Try some more example problems by yourself

Practice Problem 3: Find the missing side ‘a’ for the triangle below if (B = 40°), (c = 25), and (A = 75°).

Practice Problem 4: Find the missing side ‘b’ for the triangle below if (a = 18), (B = 40°), and (C = 100°).

Cracking the Triangle Code: A Step-by-Step Guide to Finding the Missing Side

Here is a step-by-step guide that covers all of the procedures for finding the missing side of a triangle using the methods we have discussed above.

Step 1: Determine the type of triangle by identifying its sides and angles.

Step 2: For right triangles, use the Pythagorean theorem: a² + b² = c²

Step 3: For non-right triangles use the Law of Cosines: c² = a² + b² – 2abcos(C)

Step 4: If an angle is missing, use the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

Example of Using the Guide:

Example: Find the length of the side ‘x’ for the following triangle:

Step 1: This is a non-right triangle and we have values of side a and b as well as the angle between them.

Step 2: Using the Law of Cosines: (x² = 6² + 8² – 2*6*8*cos(30°))

Step 3: Solving for x gives 9.53 (rounded to nearest hundredth).

Practice using the guide with some more example problems below.

Practice Problem 5: Find the missing side ‘b’ for the triangle below if (a = 18), (C = 25°), and (b = 21).

Practice Problem 6: Find the missing side ‘a’ for the triangle below.

Triangle Troubles? Here’s How to Find the Missing Side in No Time

If you need to solve for the missing sides of triangles quickly, here are some tips:

Tip 1: Always check if you need to use the Pythagorean Theorem before using the Law of Cosines/Sines

Tip 2: Start by labeling all sides and angles of the triangle

Tip 3: Consider which formula to use and then choose the appropriate sides and angles

Tip 4: Watch your units and use a calculator for accuracy

Practice these tips using the example problems below:

Practice Problem 7: Find the missing side ‘a’ for the triangle below:

Practice Problem 8: Find the length of the missing side ‘x’ for the following triangle:

Unlocking the Secret: Tips and Tricks to Finding the Missing Side of a Triangle

Here are some more tips and tricks to help you find the missing side of a triangle:

Tip 5: If there are two unknowns, find the smaller of the two first.

Tip 6: If there are only angles given, use them to find the missing sides by applying the Law of Sines.

Tip 7: Remember that the sum of the interior angles in a triangle is always 180 degrees.

Tip 8: In finding the missing side of a triangle, set the problem up correctly to avoid errors.

Practice using these tips with the example problems below:

Practice Problem 9: If the sides of a triangle measure 12, 16, and 22 units respectively, is the triangle acute, obtuse or right angled?

Practice Problem 10: A triangle has side lengths of 6, 8, and 10. What are the measures of its angles?

No Math Degree Required: Simple Methods to Find the Missing Side of a Triangle

If you are feeling intimidated or overwhelmed by the methods we have presented so far, here are some simplified explanations to help you find the missing side of a triangle:

Method 1: If you are given two sides of a right triangle, use the Pythagorean Theorem to find the third side.

Method 2: If you have two angles and one side of a triangle, use the sum of the angles formula (180 degrees) to find the missing angle(s). Then use the Law of Sines to find the sides.

Method 3: If you are given two sides and an included angle, then use the Law of Cosines to find the third side.

Try using these simple methods with the example problems below:

Practice Problem 11: Find the length of the missing side ‘a’ for the triangle below:

Practice Problem 12: Find the length of the missing side ‘y’ for the triangle below:

From Pythagoras to Your Homework: How to Find the Missing Side of Any Triangle

Now that you have seen and practiced various methods of finding the missing side of a triangle, it is important to understand their everyday applications. Engineers, architects, and construction workers use triangle calculations frequently in their work. Each of the methods discussed can help you solve a problem and get the job done.

Practice Problem 13: The roof of a barn is in the form of an equilateral triangle. Find the length of the rafters (i.e., the sides) if the height of the barn is 10 meters.

Practice Problem 14: An airplane is heading for a landing on a runway that is inclined at an angle of 2 degrees. If the horizontal distance from the plane to the point on the ground directly below the place is 5,000 feet, find the length of the runway to the nearest integer.

Conclusion

In conclusion, finding the missing side of a triangle may seem intimidating at first, but with practice, you will become comfortable with the different methods and ways to approach and solve problems. Remember, it is essential to understand how to find the missing side in any triangle since the skill is used in many industries and fields. So keep practicing and experimenting until you master the art of triangle calculations.