### 1. What is a 30 60 90 Triangle?

In geometry there is a presence of different varieties of triangles that can be classified into several names according to different forms of classification. Some of the triangles can be classified with the help of their side length. These are mostly considered as the equilateral, isosceles and scalene triangles. Some of the triangles can again be measured with the angles they possess. For example the right angles, the obtuse angles and the acute angles. There are even subdivisions of these classification of the triangles. The triangles can also be classified into several smaller groups but in this following section it is going to be discussed about a special kind of a right triangle which has only one right angle, or a 90 degree angle. This particular kind of triangle is known as a 30 60 90 triangle which is nothing but a right angle triangle where one angle is 90 degrees and the other two angles are consecutively 60 and 30 degrees. There is no common order of occurrence for the angles and they can occur randomly. The 30 60 90 triangle is an important triangle in geometry as it has specific relationship with the sides it possesses. it is a geometrical fact that the hypotenuse in a right angle triangle can be considered as the longest side, and in a 30 60 90 triangle the hypotenuse is exactly the opposite side of the 90 degree angle directly across. In a 30 60 90 triangle it is easier to measure any of the three sides by understanding the length of at least one side of the triangle. It can be easily seen with general geometric calculations from the basic level that the hypotenuse is double the size of the shorter side which is usually across the 30 degree angle. Again it has been found in the 30 60 90 triangle that the hypotenuse is equal to the shorter leg twice by its value and the longer leg is a cross the 60 degree angle. The measurement of the longer angle is usually measured by multiplying the shorter leg with the square root of 3. The short leg usually served as the connecting bridge between two sides of the triangle. The measurement of the longer leg to the hypotenuse all vice-a-versa can easily be obtained but if it passes through the short leg the value can be easily found. Power it is also we found that there is no direct route from the longer led to the hypotenuse or the hypotenuse to the longest leg.

### 2. Why It Works: 30 60 90 Triangle Theorem Proof

The theorem of 30 60 90 triangle states that in a particular 30 60 90 triangle the sides are in the ratio 1: 2:√3.

### The proof of the theorem would be provided as below:

It needs to be proved that the 30 60 90 triangles have the sides in the ratio of 1: 2:√3. therefore for this proof it is required to keep in mind that the smallest site one needs to be the opposite of the smallest angle 30° which is the basic rule of a 30 60 90 triangle. On the other hand decide to would have to be larger than √3 and 1: 2:√3 has the corresponding value of the 30 60 90 triangle which finds it easier to remember the sequence of 1: 2:√3.

For answering the problem regarding the 30 60 90 triangle as mentioned above, its needs to be considered that an equilateral triangle ABC is drawn. Then it needs to be specified that each of the equal angles are 60 degrees. Then a straight line is drawn bisecting it into to 30 degree angles and this line is called AD. AD is perpendicular bisector of BC.

On the other hand the triangle ABD therefore is now a 30 60 90 triangle and the triangle ADC is also 30 60 90 triangle. Therefore it implies that BD is equal to BC and BD is half of BC. On the other hand it also implies that BD is half of AB since AB is equal to BC. Therefore with this it can be proved AB: BD = 2:1.

Taking the Pythagorean Theorem. it can clearly be said that the measurement of the third side, which is AD, can be found.

AD^{2} + 1^{2} = 2^{2}

AD^{2} = 4 − 1 = √ 3

AD = 3.

Therefore it can be easily set from the above proof of the theorem that 30 60 90 triangle has its sides kept in the ratio of 1: 2:√3.

### 3. When to Use 30 60 90 Triangle Rules

Working with the 30 60 90 triangles can be easily remembered with the rules that follow the triangle. It is usually found that the rules of a 30 60 90 triangle always share the idea of having the sides as the same basic ratio of 1: 2:√3. If the angles are translated into radians then it can also be seen that the initial ratio of 1: 2:√3 still existing. The hypotenuse is regarded as the longest side of the right angle triangle which is different from the long leg and the long leg is opposite to the 60 degree angle. It is easily understandable that which is the short leg as it is half the length of the hypotenuse and the multiplication of the shorter side by the square root of 3 is usually used for finding the long leg. If somebody understands the identification of the hypotenuse then it can easily be divided by 2 to find out the short sight and multiplied answer to the square root of 3 to find the long leg. Again if some person knows which the long leg is and which is across the 60 degree angle, then the person can you really figure out that dividing this side by the square root of 3 to find the short side is easily achievable. Double of that measurement is used for identifying the hypotenuse. This can be used for finding out the basic structures and the measurements of the different length of a 30 60 90 triangle only by remembering the ratio it has about the sides. When it comes to triangle there are various varieties found that can be classified by their angle measurements and the length of the hands but the 30 60 90 triangle is special because it is consistent and predictable. The 30 60 90 triangle appears to be similar to right angle triangle but only three pieces of information required for this particular triangle to find out every measurement regarding the triangle. If the value of two of the angle measures are known and one side length is found irrespective of the length of the side, then the measurement of every kind for the triangle can be easily identified. All the information regarding the triangle can be found out only by knowing the measurement of two of the angles and one side length. Doubling the figure of the 30 60 90 triangle it can also be found that are equal triangle is formed which exactly splits into halves and therefore both the triangles that is can join to form one equilateral triangle are congruent to each other. Another case it can be found out that knowing the rules of the 30 60 90 triangle helps in saving much time and energy on identifying the multitude of different neck problems and wide variety of trigonometry difficult and geometrical problems. The understanding of the 30 60 90 triangle allows a person to solve the geometry questions that would either be impossible to solve for without the knowledge of the ratio rules they would not be considered to be what the time and effort of solving in a long way. Identifying the ratios of the special triangles the missing heights and leg length of the triangle can be easily found out with the help of the Pythagorean Theorem and the finding of the area of the triangle with the use of the missing height of the baseline information can easily be used for calculating the parameters in a quick manner.

### 4. Tips for Remembering the 30 60 90 Rules

There are also several tips that are used for identifying or remembering the 30 60 90 rules. It is identified that they are useful and handy but the information must also be remembered for identifying the rules and using it properly the rules are different for remembering the proper 30 60 90 triangle as a matter of remembering the ratio to be 1: 2:√3. Glowing the shortest side length is always opposite the shortest angle and the longest side length is always opposite the longest angle. There are few people that uses memorizing the ratio by thinking 1 2 3 is a succession that is easy to remember. However it also needs to be kept in mind that in the tips for remembering the ratio of the angles three is not just a number but it is a root over of the actual number. It can be used as a demonic wordplay as a technique to remember the longest side that it is actually 2x and not x times’ root over 3.

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