Homework 2 Special Right Triangles Calculator - Essay for you

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Homework 2 Special Right Triangles Calculator

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3 Easy Ways to Find the Length of the Hypotenuse

wiki How to Find the Length of the Hypotenuse

All right triangles have one right (90-degree) angle, and the hypotenuse is the side that is opposite or the right angle, or the longest side of the right triangle. [1] The hypotenuse is the longest side of the triangle, and it’s also very easy to find using a couple of different methods. This article will teach you how to find the length of the hypotenuse using the Pythagorean theorem when you know the length of the other two sides of the triangle. It will then teach you to recognize the hypotenuse of some special right triangles that often appear on tests. It will finally teach you to find the length of the hypotenuse using the Law of Sines when you only know the length of one side and the measure of one additional angle.

Steps Edit Method One of Three:
Using the Pythagorean Theorem Edit

Learn the Pythagorean Theorem. The Pythagorean Theorem describes the relationship between the sides of a right triangle. [2] It states that for any right triangle with sides of length a and b, and hypotenuse of length c, a 2 + b 2 = c 2. [3]

Make sure that your triangle is a right triangle. The Pythagorean Theorem only works on right triangles, and by definition only right triangles can have a hypotenuse. If your triangle contains one angle that is exactly 90 degrees, it is a right triangle and you can proceed.
  • Right angles are often notated in textbooks and on tests with a small square in the corner of the angle. This special mark means "90 degrees."

Assign variables a, b, and c to the sides of your triangle. The variable "c" will always be assigned to the hypotenuse, or longest side. Choose one of the other sides to be a, and call the other side b (it doesn't matter which is which; the math will turn out the same). Then copy the lengths of a and b into the formula, according to the following example:
  • If your triangle has sides of 3 and 4, and you have assigned letters to those sides such that a = 3 and b = 4, then you should write your equation out as: 3 2 + 4 2 = c 2 .

Find the squares of a and b. To find the square of a number, you simply multiply the number by itself, so a 2 = a x a. Find the squares of both a and b, and write them into your formula.
  • If a = 3, a 2 = 3 x 3, or 9. If b = 4, then b 2 = 4 x 4, or 16.
  • When you plug those values into your equation, it should now look like this: 9 + 16 = c 2 .

Add together the values of a 2 and b 2 . Enter this into your equation, and this will give you the value for c 2. There is only one step left to go, and you will have that hypotenuse solved!
  • In our example, 9 + 16 = 25. so you should write down 25 = c 2 .

Find the square root of c 2 . Use the square root function on your calculator (or your memory of the multiplication table) to find the square root of c 2. The answer is the length of your hypotenuse!
  • In our example, c 2 = 25. The square root of 25 is 5 (5 x 5 = 25. so Sqrt(25) = 5 ). That means c = 5. the length of our hypotenuse!
Method Two of Three:
Finding the Hypotenuse of Special Right Triangles Edit

Learn to recognize Pythagorean Triple Triangles. The side lengths of a Pythagorean triple are integers that fit the Pythagorean Theorem. These special triangles appear frequently in geometry text books and on standardized tests like the SAT and the GRE. If you memorize the first 2 Pythagorean triples, in particular, you can save yourself a lot of time on these tests because you can immediately know the hypotenuse of one of these triangles just by looking at the side lengths! [4]
  • The first Pythagorean triple is 3-4-5 (3 2 + 4 2 = 5 2. 9 + 16 = 25). When you see a right triangle with legs of length 3 and 4, you can instantly be certain that the hypotenuse will be 5 without having to do any calculations.
  • The ratio of a Pythagorean triple holds true even when the sides are multiplied by another number. For example a right triangle with legs of length 6 and 8 will have a hypotenuse of 10 (6 2 + 8 2 = 10 2. 36 + 64 = 100). The same holds true for 9-12-15. and even 1.5-2-2.5. Try the math and see for yourself!
  • The second Pythagorean triple that commonly appears on tests is 5-12-13 (5 2 + 12 2 = 13 2. 25 + 144 = 169). Also be on the lookout for multiples like 10-24-26 and 2.5-6-6.5 .

Memorize the side ratios of a 45-45-90 right triangle. A 45-45-90 right triangle has angles of 45, 45, and 90 degrees, and is also called an Isosceles Right Triangle. It occurs frequently on standardized tests, and is a very easy triangle to solve. The ratio between the sides of this triangle is 1:1:Sqrt(2). which means that the length of the legs are equal, and the length of the hypotenuse is simply the leg length multiplied by the square root of two.
  • To calculate the hypotenuse of this triangle based on the length of one of the legs, simply multiply the leg length by Sqrt(2). [5]
  • Knowing this ratio comes in especially handy when your test or homework question gives you the side lengths in terms of variables instead of integers.

Learn the side ratios of a 30-60-90 right triangle. This triangle has angle measurements of 30, 60, and 90 degrees, and occurs when you cut an equilateral triangle in half. The sides of the 30-60-90 right triangle always maintain the ratio 1:Sqrt(3):2. or x:Sqrt(3)x:2x. If you are given the length of one leg of 30-60-90 right triangle and are asked to find the hypotenuse, it is very easy to do: [6]
  • If you are given the length of the shortest leg (opposite the 30-degree angle,) simply multiply the leg length by 2 to find the length of the hypotenuse. For instance, if the length of the shortest leg is 4. you know that the hypotenuse length must be 8 .
  • If you are given the length of the longer leg (opposite the 60-degree angle,) multiply that length by 2/Sqrt(3) to find the length of the hypotenuse. For instance, if the length of the longer leg is 4. you know that the hypotenuse length must be 4.62 .

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Special Right Angle Triangles - Definition, Types & Problems

Special Right Triangles

A right angle triangle is one who has one of its angle as right angle. Trigonometry is completely based on these triangles. If one angle is a right angle then the sum of other two angles will be 90 degree. On the basis of the value of the other two angles of a right angle triangle, it can be classified as a special right triangle. The special right angle triangles are classified into three types in geometry. The classification of the special right angle triangles is based on their angles as well as their sides. These triangles are 45-45-90, 30-60-90 and 3-4-5 triangles. The first two classifications are on the basis of measure of angles and the third classification is on the basis of measure of sides. The name of special right angle triangles are isosceles right triangle. scalene right triangle and 3-4-5 triangle.

Types of Special Right Angle Triangles

Let us have a detailed explanation of special right triangles.

Isosceles right triangle. In a right angled triangle if two sides and two angles are equal then it is known as isosceles right angled triangle. The ratio of sides is $n:n:n\sqrt2$. The angles are 45-45-90.


Scalene right triangles. A right angled triangle who has all three angles are different, one angle is 30 degrees, second angle is 60 degrees, and third angle is right angle is know as scalene right triangle. The ratio of these angles is 1:2:3. The ratio of sides is $n:n\sqrt 3: 2n$.

3-4-5 triangle: If in a right angle triangle, the sides of the triangle is always in a fixed ratio of 3:4:5 then it is called a 3-4-5 triangle. Conversely, it can be said that if the ratio of sides of a triangle is given to be 3:4:5 then it will be a right angled triangle.

Below are the examples based on types of special right angle triangles -

In a Isosceles right triangles two side length is 7m. Find the largest side.


From Phythagorean Theorem we know that

so the length of largest side is 7$\sqrt2$ m.

Also,it is proved that the ratio of the side $7:7:7\sqrt2$ is equal to $n:n:n\sqrt2$.

Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 7 inches and 7$\sqrt3$ inches. Identify the type of triangle. Is it a special right triangle?


From Phythagorean Theorem we know that

$AC^2$ = $(7\sqrt3)^2$ + $7^2$

so the length of hypotenuse side is 14 inches.

As the ratio of sides is $7:14:7\sqrt 3 = n:2n:n\sqrt 3$, it can be classified as special right triangle, that is, scalene right triangle.

Find the length of the largest side of a right triangle if the lengths of the other two sides are 6 inches and 8 inches. Is it a special right triangle?


From Phythagorean Theorem we know that

Hence, the length of hypotenuse side is 10 inches.

It can be seen that the ratio of the side 6:8:10 is equal to 3n:4n:5n. Hence, it is a 3-4-5 triangle which is a special right triangle.

The Best Right Triangle Calculator

Why to Use a Right Triangle Calculator

There are many students who agree that geometry is one of the trickiest subjects so that you may have to face certain challenges and complications when studying it. Unfortunately, some students end up with low grades because it’s really not as simple as it may seem. If you worry about geometry lessons and assignments, the good news is that you can use a variety of helpful tools, such as a right triangle calculator and the quality services of experienced freelancers who are easily available online.

If you’re assigned to learn more about right triangles, you should understand that they are those shapes that have 1 right angle (measured in 90 degrees). Keep in mind that the existing relation between angles and sides in a right triangle is one of the key fundamentals when it comes to trigonometry. There are certain terms and definitions that should be mastered to get a better understanding of what this shape is all about.

What is a hypotenuse? It’s a side opposite to the right angle, and the sides adjacent to it are legs. Why do you need to use a right triangle calculator? Sometimes, you may not have enough time and knowledge to solve problems, and this is when this helpful and easy to use tool may come in handy. If the lengths of all triangle sides are integers, this shape is called a Pythagorean triangle (all of its side lengths are called a Pythagorean triple). Do you need to solve specific problems to do your geometry homework successfully? Browse the Internet for a helpful and easy to use right triangle calculator, and the only step that should be taken to calculate an unknown angle, side, or area is entering 2 known angles or sides. The best part is that you can find step-by-step explanations and guidelines given for every calculator of this kind.

The Definition of Special Right Triangles

When it comes to the values of trigonometric functions, they can be evaluated for given angles with the help of a right triangle with special angles. For example, take a look at 30-60-90 triangles that are often used by students to evaluate trigonometric functions. Don’t forget about 45-45-90 triangles because you may need them when evaluating such functions.

What is a special right triangle? Basically, it’s a triangle that has some regular features that make different calculations easier. You can use simple formulas or an online right triangle calculator to come up with the right answer. If right triangles have the angles measured as 45, 45, and 90 degrees, they are called angle-based. Focus on side based triangles because the lengths of their sides form a ratio of 3-4-5 or other numbers, including the golden ratio. If you know the side ratios or angle relationships of such triangles, it becomes quite fast and easy to calculate different lengths in given geometry problems without using more advanced methods or tools.

Keep in mind that angle-based right triangles are always specified by the existing relationships of angles that form them. If their sides are integer lengths are called Pythagorean triples and they have those angles that can’t be rational numbers. They are quite useful because they are easy to remember and the multiples of their sides produce the same relationship.

The Basic Properties of Right Triangles

Nowadays, right triangles are widely used in different branches of mathematics. Take a look at specific trigonometry concepts because they are related to their properties. As a geometry student, you should be familiar with the Pythagoras Theorem, which determines the relationship between 3 sides of such triangles.

A hypotenuse is the triangle side opposite to its right angle, and it’s always the longest side. Other 2 sides can’t be a hypotenuse, and they always make a right angle. There are other properties that you should learn before you start using a right triangle calculator to solve geometry problems.

It’s worth mentioning that right triangles can be isosceles if their 2 sides that form a right angle are equal in their lengths. However, such shapes can’t be equilateral because their hypotenuse must be longer than other 2 sides. How to construct a right triangle? Your professors may ask you to complete this academic task in your assignment so that you should be well-prepared. The good news is that it’s easy to construct this shape if you have a straightedge or a compass given different combinations of angles and sides. If you have any difficulties when solving relevant problems, feel free to use an online right triangle calculator to save your time and get answers.

Isosceles and Equilateral Triangles

As you already know, isosceles triangles have 2 sides of the same length, but if all 3 sides are equal, such triangles are called equilateral. This means that all equilateral triangles have all important properties of the isosceles ones. What are their basic properties?

Any unequal side of isosceles triangles is called their base, and it always remains equal. If one of their angles is right, these shapes are called right isosceles triangles and their altitude is a perpendicular distance from their topmost vertex to a base.

When it comes to equilateral triangles, all of their sides are congruent or equal, just like their angles. They are called the specific cases of regular polygons so that all properties and facts associated with such polygons are applied to equilateral triangles. Keep in mind that all of their angles are always measured 60 degrees. If you need to calculate their area, it’s possible to use a traditional method, but a specific formula can be applied in certain cases. If your geometry homework includes questions and problems related to right triangles, remember that you have a simple tool that can help, and it’s a right triangle calculator.

Simple and Effective Geometry Tips

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You need to separate your academic tasks to simplify the entire process of doing your geometry homework. It’s necessary to determine their urgency, importance, and complexity. For instance, some of them can wait, while others must be done as soon as possible. Start with the assignments that require urgent solutions and take a lot of time to complete.

Many students make one common mistake because they start feeling quite frustrated only because they have no time. Remember that being irritated and puzzled won’t help you handle your homework properly, so calm down before getting started. Take into account that there are many simple and effective tools that can help you, including a right triangle calculator.

Another helpful idea is to organize your working space effectively. Look for a place where you won’t feel distracted and prepare everything you need. It’s advisable to get rid of all possible distractions in advance and determine where you want to start. Some geometry students decide to start with the hardest task of their homework when they still have a lot of energy and time, while others feel motivated after completing the easiest academic tasks before concentrating on the most complicated ones. You should choose your own path according to personal preferences and your learning style. Such incredible tools as a right triangle calculator will always help you, regardless of the choice you make.

If you want to save more time, avoiding procrastination is one of the most effective methods to achieve this goal. You should understand that postponing your academic tasks is a bad idea because the later you start, the less time you will get to relax. Make sure that you start doing your math homework as early as you can. Sometimes, you may need a few minutes to unwind and feel inspired to solve multiple geometry problems, but ensure that this relaxation period won’t turn into laziness hours. When completing your right triangle assignments, take short breaks because your brain needs them for improved productivity.

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