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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.

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]

- 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."

- 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*.

- 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*.

- In our example,
*9 + 16 = 25*. so you should write down*25 = c 2*.

- 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!

Finding the Hypotenuse of Special Right Triangles Edit

- 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*.

- 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.

- 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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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.*

Let us have a detailed explanation of special right triangles.

I*sosceles 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.

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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.

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.

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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.

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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.

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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.

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