How to Find the Missing Side of a Triangle: The Ultimate Guide

Introduction

Triangles are one of the most fundamental geometrical shapes, and learning how to work with them is essential for many fields of study, from engineering and architecture to physics and mathematics. One common problem people face when working with triangles is finding the missing side of a triangle. In this article, we will explore various methods to solve this problem and provide helpful tips and tricks to identify the right method for the given information.

Using Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. Its formula is a² + b² = c², where ‘a’ and ‘b’ are the lengths of the legs of the right triangle, and ‘c’ is the length of the hypotenuse.

To use the Pythagorean Theorem to find the missing side of a triangle, identify the two known sides of the right triangle and plug them into the formula to solve for the unknown side. For example, if we know that one leg of a right triangle is 3 cm long, and the hypotenuse is 5 cm long, we can use the Pythagorean Theorem to find the length of the other leg:

a² + b² = c²

3² + b² = 5²

9 + b² = 25

b² = 16

b = 4

So the missing side of the triangle is 4 cm long.

It’s important to remember that the Pythagorean Theorem only works for right triangles. To identify a right triangle, look for a triangle where one angle measures 90 degrees.

Using Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. There are three primary trigonometric ratios: sine, cosine, and tangent, which can be used to find the missing side of a triangle.

The sine ratio (opposite/hypotenuse), cosine ratio (adjacent/hypotenuse), and tangent ratio (opposite/adjacent) are defined as the ratios of the lengths of the sides of a right triangle relative to its acute angles.

Suppose we have a right triangle where one angle measures 30 degrees, and we know the length of one of its sides and the measure of another angle. To find the length of the missing side, we can use the sine, cosine, or tangent ratios, depending on the given information. For example:

In a triangle ABC, where angle A = 30° and AB = 4 cm. Find BC.

30-degree-triangle

Using the sine ratio:

sin(30°) = opposite/hypotenuse

sin(30°) = BC/4

BC = 4sin(30°)

BC = 2 cm

To use trigonometry effectively, it’s essential to know which ratio to use for which information. The mnemonic SOHCAHTOA can help you remember which ratio to use:

Sine: Opposite/Hypotenuse
Cosine: Adjacent/Hypotenuse
Tangent: Opposite/Adjacent

Using Similar Triangles

Similar triangles are triangles whose corresponding angles are congruent, and their corresponding sides are proportional. They have the same shape but different sizes.

One of the most practical applications of similar triangles is to use them to find the missing side of a triangle. Suppose we have two similar triangles, and we know the length of one corresponding side of each triangle. In that case, we can set up and solve the proportion of the corresponding sides to find the length of the unknown side. For example:

In a triangle ABC, where the angle A is equal to angle C and AB = 5 cm, we draw a perpendicular from B on AC and mark it as BD. If BD = 3 cm, find BC.

similar-triangles

Since triangles ABC and BDC are similar, we have:

BC/AB = BD/BC

BC² = AB x BD

BC² = 5 x 3

BC = √15 cm

To identify similar triangles, look for triangles that have the same shape but different sizes. They usually have two or more corresponding angles that are congruent.

Special Triangles

Special triangles are triangles with specific angles and side relationships that make them unique. The two most common special triangles are the 30-60-90 triangle and the 45-45-90 triangle.

The 30-60-90 triangle has angles of 30°, 60°, and 90° and is characterized by the relationship between its sides, where the length of the longer leg is the square root of 3 times the length of the shorter leg, and the length of the hypotenuse is twice the length of the shorter leg.

The 45-45-90 triangle has angles of 45°, 45°, and 90° and is characterized by the relationship between its sides, where all sides are congruent, and the length of each side is equal to the square root of 2 times the length of one of the legs. Using these unique properties, we can find the missing side of these triangles with ease.

Using Law of Cosines and Law of Sines

The Law of Cosines and the Law of Sines are two important formulas used to solve triangles that are not necessarily right triangles.

The Law of Cosines states that the square of the length of one side of a triangle equals the sum of the squares of the other two sides minus twice the product of the lengths of those sides and the cosine of the included angle. The equation for the theorem is a² = b² + c² – 2bc cos(A), where ‘a’ is the length of the opposite side, and ‘b’ and ‘c’ are the lengths of the other two sides, and ‘A’ is the included angle.

The Law of Sines states that the ratios of the lengths of the sides of a triangle are proportional to the sines of their opposite angles. The equation for the theorem is a/sin(A) = b/sin(B) = c/sin(C), where ‘a’, ‘b’, and ‘c’ are the lengths of the sides, and ‘A’, ‘B’, and ‘C’ are the opposite angles.

To use these laws to find the missing side of a triangle, apply the appropriate formula based on the information given. For example:

In triangle ABC, where AB = 8 cm, AC = 10 cm, and angle A = 60°, find the length of BC.

Using the Law of Cosines:

a² = b² + c² – 2bc cos(A)

BC² = 8² + 10² – 2(8)(10) cos(60°)

BC = √174 cm

Using the Law of Sines:

a/sin(A) = b/sin(B) = c/sin(C)

BC/sin(60°) = 8/sin(B) = 10/sin(C)

BC = 8(sin(60°)/sin(B))

BC = 6.928 cm

Choosing the Best Method

Choosing the best method to find the missing side of a triangle can save time and effort. Generally, identifying the type of triangle (right, acute, or obtuse) and the given information is essential in selecting the method.

For example, if you know that you’re dealing with a right triangle, start by using the Pythagorean Theorem. If you know two sides and an included angle, use the Law of Cosines. If you know two angles and one side, use the Law of Sines. If the triangle is a special triangle, use the given side ratios to find the missing side. And if two triangles are similar, set up and solve a proportion to find the missing side.

Conclusion

Learning how to find the missing side of a triangle is a fundamental skill that can come in handy in many fields of study. Whether you’re working with right triangles, trigonometry, similar triangles, special triangles, or the Laws of Cosines and Sines, knowing when and how to use each method will make your work more efficient and accurate.

With the tips and tricks provided in this article, you’re now equipped to tackle the challenge of finding the missing side of a triangle with confidence. Keep practicing, and soon you’ll master this essential skill.