## Introduction

Triangles show up everywhere in our lives – in architecture, science, and nature. In many cases, we might know some sides of a triangle and need to find the missing one. While this might seem like a daunting task for those who once struggled with math, there’s no need to panic. Finding the missing side of a triangle is not rocket science if you know the right methods. That is what this article is all about; we want to teach you the most straightforward ways of finding the missing sides of triangles so that you can apply them to solve real-world problems.

This article will cover the basics of triangle solving, various techniques for finding missing sides of triangles, and their practical applications.

## 5 Simple Steps to Find the Missing Side of a Triangle

Before we delve into the five simple steps of finding the missing side of a triangle, we must first understand the three types of triangles:

– Scalene triangle: A triangle with no equal sides

– Isosceles triangle: A triangle with two equal sides

– Equilateral triangle: A triangle with three equal sides

For all triangle types, these five steps should work:

1. Measure or find the length of at least one side of the triangle.

2. Look for angles that you know or can solve using other methods.

3. Identify the trigonometric functions you can use to calculate the missing side of the triangle.

4. Set up an equation using the appropriate function.

5. Solve the equation for the missing side.

Here’s an example:

Suppose we have an isosceles triangle in which the two equal sides are 6cm long. We want to find the length of the base.

Step 1: Measure one of the equal sides, which is 6cm.

Step 2: An isosceles triangle has two equal angles opposite the two equal sides. Using this information, we know that the angle opposite the base is (180 – 2x angle A) /2.

Step 3: The tangent function will be useful in this problem.

Step 4: We can set up our equation as follows:

tan(A) = Opposite / Adjacent = base side / 6cm

Step 5: Cross-multiply to solve for the base length:

tan(A) x 6cm = base length

You can use a scientific calculator to find the value of tan(A) or look it up on a table.

## Trigonometry 101: Solving for the Missing Side of a Triangle

Trigonometry is a branch of mathematics that deals with the relationship between the sides and angles of a triangle. It provides a useful tool for finding missing sides in any triangle.

To use trigonometry to solve sides, we’ll focus on sin, cos, and tan, which are known as trigonometric ratios. These ratios are calculated using the following formulas:

– Sine: sinθ = Opposite / Hypotenuse

– Cosine: cosθ = Adjacent / Hypotenuse

– Tangent: tanθ = Opposite / Adjacent

Once you have one side and two angles, you can use the trigonometric functions to find the missing side of the triangle. Remember to choose the trigonometric function based on which sides in the triangle you already know.

Let’s see an example:

Suppose we have a triangle with sides of 8 and 10 units and an included angle of 60 degrees.

Using the cosine function, we can solve for the missing side (x):

cos 60 degrees = x / 10

x = 10 x cos60 degrees ≈ 5 units.

## The Pythagorean Theorem: A Guide to Finding Missing Triangle Sides

The Pythagorean theorem can be used to find missing sides in right triangles. A right triangle has one 90-degree angle and two other angles that add up to 90 degrees. The theorem is as follows:

a² + b² = c²

In this formula, a and b are the lengths of the two sides that meet at the right angle, while c is the length of the hypotenuse.

To find a side of a right triangle using the Pythagorean theorem, we need two sides already known, one of which must be the hypotenuse.

For instance, suppose we have a right triangle with legs measuring 3cm and 4cm. We want to find the length of the hypotenuse.

Using the Pythagorean theorem, we can write:

c² = a² + b²

c² = 3cm² + 4cm² = 9 + 16 = 25

c = √25cm = 5cm.

Therefore, the length of the hypotenuse is 5cm.

## How to Use Similar Triangles to Find the Missing Side

Similar triangles are two triangles with the same shape but different sizes. The ratio of the lengths of the corresponding sides of two similar triangles is called the scale factor.

Using similar triangles, we can find the length of a corresponding side when we know the length of at least one side and the ratio of the corresponding sides.

For example, suppose we have two similar right triangles in which the sides are in proportion to each other, and we know the ratio of the sides is 2:3.

Suppose one triangle has a hypotenuse of 10cm. We can use the ratio of the corresponding sides and write:

2/3 = x/10

Now we can cross-multiply to solve for x:

2 x 10 = 3x

x = (2×10)/3

= 6.67cm.

Therefore, the length of the hypotenuse of the second triangle is 6.67cm.

## Geometry Hacks: Finding the Missing Side of a Right Triangle

Sometimes you need to find triangle sides quickly and easily, and you don’t have the time or tools to use formal methods. Here are some tricks that may help you find missing sides in a right triangle:

– 3, 4, 5 rule: If two legs of a right triangle measure 3 and 4 units, respectively, then the hypotenuse measures 5 units. Similarly, if two legs of a right triangle measure 6 and 8 units, respectively, the hypotenuse measures 10 units.

– 45-45-90 rule: In an isosceles right triangle, if one leg has a length of 3, then the hypotenuse and the other leg have lengths of 3√2 units.

– 30-60-90 rule: In a right triangle where one angle measures 30 degrees, one leg’s length is half that of the hypotenuse, and the other leg’s length is √3/2 times that of the hypotenuse.

## No Calculator Needed: Mental Math Tricks for Finding the Missing Triangle Side

Mental math tricks come in handy when you don’t have a calculator handy. Here are some fast methods:

– Multiplying by 2 and 5: Doubling a side’s length is equivalent to multiplying by 2, while multiplying by 5 is the same as adding a zero.

– Longest side: When given angles and two sides, the longest side will always be opposite the largest angle. Conversely, the smallest side will be opposite the smallest angle.

– 3-4-5 rule: If one side has a length of 3, and the other has a length of 4 units, the missing side must have a length of 5 units. This rule works for larger numbers too, so long as the ratio of sides remains the same.

## Common Applications of Triangle Solving: Finding Missing Sides for Real Problems

Finding missing sides is a crucial step in a wide range of real-world applications. Engineers use it to determine the correct distance between objects, calculate the size of gears for accurate gear ratios, and create accurate 3D models of buildings. Architects use it to draw building plans and elevations, ensuring that each structure’s dimensions are correct.

## Conclusion

There are several ways to find missing sides of triangles easily, quickly, and without stress. By practicing frequently, you can get better at using these methods and applying them to real-world problems. Remember to choose a method based on the type of triangle you are dealing with, and always use the information you already know to solve for the missing information.