The 30 60 90 triangle is a special type of triangle that examines 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 an important role in high school, colleges and even SAT exams. There are different types of triangles that students 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, their computation is generally limited to boundary triangle computations.
A particular triangle that can explore much more than the triangle measured above is the 30 60 90 triangles. Geometrically, the triangle 30 60 90 is basically aright triangle
However, it is not just any right angle. The right angle must have three angels divided as 300, 600and 90. This article will provide a comprehensive understanding of triangles 30, 60, and 90 so students can understand and easily score whenever they need to manipulate them during the SAT.
We will consider the implications 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 denoted as 300, 600and 900.These angles are constant and will always have its three angles constant as 300, 600and 900. There is no specific 30 60 90 triangle formula as it is more about logic and theory.
Now let's look at our right angle in the first section.
How to find the longest side of a triangle: triangle inequality theorem
In math, it's important to understand that the side opposite the largest angle of a triangle is always the largest side. Also in our investigation oftriangle inequality theoremwe were able to show consistently that the longest side is always opposite the longest angle.
This is easy to show geometrically. In the following we consider a triangle
According to the right angle philosophy, the lower angle of side a is always 90°0. It is the only angle that is represented with a square instead of a curve. Now assume that side b, which is the hypotenuse, is always the longest side of any right angle. This is because the other two angles can never reach 900
Until now, students may not have understood why side b was always the longest side. That's because it's the side directly opposite side a where the biggest 90 are0angle based. The second longest side is side a when 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 takes the position of side a instead 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 keep in mind to get the right answer
Nr. 1:We already know that the angle is 900above. Now the 60th0and 300must be one of the two angles. For the 30th0600and 900Triangle is always the basic position of angles
N0 2:From the triangle above, we now know where each angle should be placed in our triangle.
Properties of the 30 60 and 90 triangle
Now there are certain things we need to know about the triangle above the triangle.
Side c is the shortest angle of the triangle and that's because it's the side opposite the shortest angle of 300. Side a follows as this is the side opposite the 600. Side b is the largest side because it is the side opposite the largest angle 900.
As known from right angles, side b is the hypotenuse, side a is the opposite side, and side c is the adjacent.
The length of the adjacent side c is given as x
The length of the hypotenuse, which is side b, is given as 2x
That is, the triangle 30 60 90 is actually
Knowing the right triangle rules is very important as it will help students boycott too many calculations and get their answers quite easy when they have to deal with this type of triangle.
For each triangle, the total angle is also 1800. Whether it's an equilateral triangle or even a right triangle, part of the total angle is 1800. This is why a right triangle already has 900at the intersection of the opposite and the adjacent there are only 900available for the other two angles to share. In the triangle 30 60 90 are the remaining 900shares 600and 300.
This property actually helps people find the angles of a right angle as long as one is the non-900angle is present.
It is a must that the adjacent side is always the shortest side. While it depends on the size of the triangles which side is the shortest of all three, students must make a conscious effort to fix 300at the intersection between the opposite hypotenuse, which is the same as that directly opposite the adjacent one. When we celebrate the 30th0600and 900in the form below;
Then the shortest side is automatically the opposite side and the second shortest side is the adjacent one. This type of triangle cannot be considered a 30 60 90 triangle.
The 30 60 and 90 triangle uses the Pythagorean theorem like any type of right angle. The formula of Pythagoras' theorem states that the square of a hypotenuse is equal to the sum of the squares of its adjacent and opposite hypotenuses.
It is actually given as;
hyp2= adj2+ opp2
When is the right time to apply the 30-60-90 triangle philosophy?
In SAT it is very common to deal with the triangle 30 60 90. Although it may not be entirely clear that this is the triangle 30 60 90, when students are presented with such angles it is fairly easy to tell based on the properties discussed.
- If there is a orthogonal calculation where one angle is available, and it's actually 300or 600, then such a calculation is definitely a 30 60 90 triangle problem. This is because at the moment one of the two angles is 300and 600are present, it is only logical that the other is the missing piece as a right angle of 900is always constant.
- Given an angle and two sides, or given two sides and an angle, we can also solve the right triangle using the Pythagorean theorem.
When you talk about itTrigonometryProblems, let's basically talk about the sine cosine and tangent of trigonometry. There is a relationship between the 30 60 90 triangle and the trigonometry function.
Now after trigonometry,
- Sine, when expressed as an angle, will be
Sin (ϴ of intersection of opposite and hypotenuse). For this reason, it is called SOH, where S is the sine, O means the opposite, and H means the hypotenuse
- For cosine; It is
Cos (ϴ of the intersection of adjacent and hypotenuse). For this reason, it is referred to as CAH, where C is the cosine, A means adjacent, and H means hypotenuse
- For tangent; It is
Tan (ϴ of intersection of opposite and adjacent). For this reason, it is referred to as TOA, where T is the tan, O means opposite, and A means adjacent
Now, given only the angles of a 30 60 90 triangle, we can find the sides in terms of the three SOH CAH TOA of trigonometry. The answer for each of the trigonometric angles is the length of the side opposite them. How to find the length of a side using trigonometry
We will look at different examples of the 30 60 90 triangle
How do you calculate the 30 60 90 triangle?
The next few examples will explore how to love the SAT FAQs about this particular triangle.
Consider the triangle below and find its hypotenuse
The triangle is clearly a 30 60 90 triangle since two angles are provided, namely 30 and 90. Of course, 60 is the missing angle.
Now the task is to find the length of the hypotenuse.
Apparently, only the opposite has a known length, which is given as 56 cm
So how do we find the hypotenuse?
Now remember that for this kind of angle;
Adjusted length = x
Length of hypotenuse = 2x
With the above
Since we now know x, this means that we automatically know our neighbors.
Length of hypotenuse = 2x = 2(34.064) = 68.128 cm
This will give our triangle the shape.
Find theScopeof the lower triangle.
There is no need to panic when seeing the hypotenuse in roots. Students must follow the same process detailed in this article to get their answers.
We now have all sides of our triangle ready. The next step is to calculate the perimeter.
Perimeter of triangle = sum of all sides
We can calculate thatAreaof the triangle, once we found the sides, we calculated the perimeter using the well-established formulas based on the logic of the right triangle 30 60 90. The stumbling block is simply finding the sides, and the rest is simply using established triangle formulas.
This next example looks at a very common question that comes up on SAT to help students get better information about what they may encounter when they are grappling with the 30 60 90 triangle.
Consider triangle ABC. How long is AD?
A close look at the triangle above shows that we have two 30 60 90 triangles.
From the left, the first 30 60 90 triangles are indicated as BAD, while the second on the right is indicated ad BCD.
Both 30 60 90 triangles are equal to each other and as such the length of one is automatically translated to the length of the other.
Since we are looking for AD which is actually the neighbor of the first 30 60 90 triangle given as ABD we can simply find the length dc of the second triangle to know. Longitude DC is the neighbor of the second 30 50 90 triangle.
We need to use the neighbor line of the second triangle since its hypotenuse is readily available.
Clear 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, equals 15/2 = 7.5 cm
The 30 60 90 triangle is actually quite easy to calculate when faced with a challenge. All students need to do is understand its philosophies and characteristics. This piece has comprehensively outlined everything about the 30 60 90 triangle and everything that makes it up, and as you can see it's pretty easy to understand.
All of the examples considered in this article are inspired by probable SAT questions and are among the most difficult 30 60 90 triangle challenges students will ever face. Each has been thoroughly explained to enable students to understand and solve similar related questions.