The 30 60 90 Triangle is a special type of triangle that explore the properties of triangles, right angles and the triangle inequality theorem for answers.
The triangle is one of the most studied shapes in mathematics. It plays a major role in High school, colleges and even SAT exams. There are different types of triangles that students will have to deal with. These triangles include the Equilateral triangle, Isosceles triangle, Acute Triangle and Obtuse Triangle. All of these angles have their uniqueness. However, computing them is generally restrictive to borderline triangle calculations.
A special triangle that can explore much more than the above triangle measured is the 30 60 90 triangles. Geometrically, the 30 60 90 triangle is basically a right-angle triangle
However, it is not just any type of right angle. The right angle must have three angels that are shared as 300, 600 and 90. This article will provide a comprehensive understanding of the 30 60, and 90 triangles so that students can understand it and easily score high whenever they have to handle them during SAT.
We will consider the impact of the Triangle Inequality Theorem and the Right angle property to arrive at our answer easily.
What is a 30-60-90 Triangle?
The 30 60 and 90 Triangle is a special right angle with three angles given as 300, 600 and 900. This angles are constant and will always have his three angles consistently as 300, 600 and 900. There is no dedicated 30 60 90 triangle formula as it is more of logic and theory.
Now let us consider our right angle in the first section.
How to Know the Longest side of a Triangle: Triangle Inequality Theorem
In Mathematics, it is important to understand that the side opposite the largest angle of a triangle is always the largest side. Even in our study of the triangle inequality theorem, we could consistently show that the largest side always lies opposite against the longest angle.
Geometrically, it is easy to show this. We will consider a triangle below
Based on the philosophy of the right angle, the bottom angle of the side a will always be 900. It is the only angle represented with a square instead of a curve. Now, side b, which is the hypotenuse, is always assumed to be the longest side of any right angle. This is because the other two angles can never get up to 900
Prior to now, students may not have known why the side b was always the longest side. It is because it is the side directly opposite side a where the biggest 900 angle is based. The second longest side will be side a if the angle between the intersection of b and c (it is the angle opposite side a) is the second largest. Side b in the same vein will instead take the position of side a if the angle between the intersection of a and b (it is the angle opposite side c) is the second largest.
The above is the way to know which side is the longest side of a triangle.
There are some points we need to note to get the right answer
No 1: We already know the square angle is 900 above. Now the 600 and 300 will have to be any of the two angles. For the 300 600 and 900 triangle, the basic position of the angles is always
N0 2: From the above triangle, we now know where each angles should be placed in our triangle.
Properties of the 30 60 and 90 Triangle
Now there are certain things that we need to know about the above the triangle.
Side c is the shortest angle of the triangle and this is because it is the side opposite the shortest angle of 300. Side a follows as it is the side opposite the 600. Side b is the largest side as it is the side opposite the largest angle 900.
As is known about the right angle, the side b is the hypotenuse, side a is the opposite while side c is the adjacent.
The length of the adjacent which is side c is given as x
The length of the hypotenuse which is side b is given as 2x
That is, the 30 60 90 triangle is actually
Knowing the rules of the right angle triangle is very important because it helps student boycott too many calculations and gets their answers quite easily whenever they have to deal with this kind of triangle.
For every triangle, the entire angle is also 1800. Whether it is an equilateral triangle or even the right angle triangle, some of the entire angle is 1800. This is why since a right-angle triangle already has 900 at the intersection of the opposite and the adjacent, there is only 900 available for the two other angles to share. In the 30 60 90 triangle, the remaining 900 is shared between 600 and 300.
This property actually helps people find the angles of a right angle as long as one of the non-900 angle is available.
It is a must that the adjacent will always be the shortest side. While It depends on the size of the triangles to tell us which side is the shortest among all three, students must make a conscious effort to set 300 at the intersection between the opposite the hypotenuse, which is the same as directly opposite the adjacent. If we have the 300 600 and 900 in the form below;
Then the shortest side will automatically be the opposite side and the second shortest side will be the adjacent. This type of triangle cannot be considered as a 30 60 90 triangle.
The 30 60 and 90 Triangle uses the Pythagoras theorem just like any type of right angle. The Pythagoras theorem formula states that the square of a hypotenuse equals the sum of the squares of its adjacent and opposite.
It is actually given as;
hyp2 = adj2 + opp2
When is the Right Time to Use the 30 60 90 Triangle Philosophy?
In SAT, it is very common to deal with the 30 60 90 triangle. While it may not be totally spelt out that it is the 30 60 90 triangle, based on the properties discussed, it is quite easy to tell when students are presented with such angles.
- If there is a right angle calculation in which there is one angle available, and it is actually 300 or 600, then such computation is definitely a 30 60 90 Triangle problem. This is because the moment any of the two angles 300 and 600 are available, it is only logical that the other one is the missing piece as the right angle of 900 is always constant.
- If there are one angle and two sides available or there are two sides and one angle, we can also solve the right angle triangle with the Pythagoras theorem.
When talking about trigonometry problems, we are basically talking about the Sine Cosine and Tangent of trigonometry. There is a relationship between the 30 60 90 Triangle and the trigonometry function.
Now according to trigonometry,
- Sine when expressed in terms of angle, will be
Sin (ϴ of the intersection of opposite and hypotenuse). This is why it is termed as SOH where S is the sine, O means Opposite, and H Means Hypotenuse
- For cosine; it is
Cos (ϴ of the intersection of adjacent and hypotenuse). This is why it is termed as CAH where C is the cosine, A means Adjacent and H Means Hypotenuse
- For Tangent; it is
Tan (ϴ of the intersection of opposite and adjacent). This is why it is termed as TOA where T is the tan, O means Opposite, and A Means Adjacent
Now, if we are given just the angles of a 30 60 90 triangle, we can find the sides with respect to the three SOH CAH TOA of Trigonometry. The answer for each of the trigonometric angles will be the length of the side opposite them. That is how to find the length of the sides using trigonometry
We will consider different examples of the 30 60 90 Triangle
How to Calculate the 30 60 90 Triangle?
The next set of examples will consider how to love the commonly asked SAT questions concerning this special triangle.
Consider the triangle below and find its hypotenuse
The triangle is clearly a 30 60 90 triangle since two angles are made available i.e 30 and 90. Naturally, 60 is the missing angle.
Now the aim is to find the hypotenuse length.
Clearly, only the opposite has a known length which is given as 56 cm
So how do we find the hypotenuse?
Now recall that for this type of angle;
Adj length = x
Hypotenuse length = 2x
With the above,
Since we know x now, it means we automatically know our adjacent.
Hypotenuse length = 2x = 2(34.064) = 68.128 cm
This will make our triangle take the form.
Find the perimeter of the triangle below.
There is no need to panic seeing the hypotenuse in roots. Students need to follow the same process that has been extensively explained throughout this article to get their answers.
We now have all of the sides of our triangle completed. The next step is to compute the perimeter.
Perimeter of triangle = the sum of all sides
We can calculate the area of the triangle the same we have calculated the perimeter once we have found the sides using the established formulas built on the logic of the 30 60 90 right angle triangle. The bone of contention is simply to find the sides, and the rest is just to use simple established Triangle formulas.
This next example will consider a very common question that comes up on SAT to help students get better information on what they may encounter whenever they have to deal with the 30 60 90 Triangle.
Consider the triangle ABC. What is the length of AD?
A good look at the above triangle shows that we have two 30 60 90 triangles.
From the left side, the first 30 60 90 triangles are given as BAD, while the second one is on the right side, which is given ad BCD.
Both 30 60 90 triangles are equal to each other, and as such, whatever the length of one will automatically translate to the length of the other.
Since we are looking for AD, which is actually the adjacent of the first 30 60 90 triangle given as ABD, we can simply find the length dc of the second triangle to know it. Length DC is the adjacent of the second 30 50 90 triangle.
We have to use the adjacent of the second triangle because its hypotenuse is readily available.
Clearly 15 cm = 2x
X = 15/2
As such DC = 15/2
Since DC = 15/2 and DC = AD
Then AD, which is the main length we want to find, is equal to 15/2 = 7.5 cm
The 30 60 90 triangle is actually quite easy to compute when any challenge is brought around it. All that students need to do is just to understand its philosophies and properties. This piece comprehensively outlined everything about the 30 60 90 triangle and all that it stands, and as can be seen, it is quite easy to understand.
All the examples considered in this article are inspired by likely SAT questions and ae among the most difficult 30 60 90 triangles challenges that students will ever face. Each one was comprehensively explained so that students can understand and solve similar related questions.