Hypotenuse

The Hypotenuse is one of those words that most people think they know the definition of, but in fact, don't. First off, it has nothing to do with long things such as pencils or telescopes. Specifically, under geometry, the Hypotenuse is typically the side of a right triangle that's just opposite the 90-degree angle. If you do not know what a right angle is in a triangle? The right angle in a triangle is just one that falls opposite to the Hypotenuse. On the other hand, a right angle occurs four times in squares and rectangles.

An interesting fact about the Hypotenuse is that it's always the longest side of a right triangle. Furthermore, in a right triangle, the square of the length of the Hypotenuse is equal to the sum of the squares of the other two sides. This theorem is known as Pythagoras' theorem. 

Applications of the Hypotenuse:

There are a few real-world applications that use the Hypotenuse. One example is in navigation. Mariners often use triangulation to calculate their location by measuring the angles of two known points and then using the Hypotenuse to calculate the distance between them. This is because the Hypotenuse is the only side that doesn't change when you measure it in a triangle.

Another application of the Hypotenuse is in electrical engineering. In particular, AC power transmission lines use triangular bundles of three wires to help maintain balanced voltage and current levels. The Hypotenuse of these triangles forms the neutral wire.

Hypotenuse is one of those words that most people think they know the definition of, but in fact don't. First off, it has nothing to do with long things such as pencils or telescopes.

Find The Hypotenuse, Opposite, and Adjacent

In a right triangle, the "hypotenuse" is the longest side, an "opposite" side is the line on the other side of a given angle that does not form the angle of choice, and an "adjacent" side is subsequent to a given angle that forms the angle of choice. 

Find the Length of the Hypotenuse

The Power of Pythagorean Theorem: In order to calculate the length of the Hypotenuse of a right triangle, you can use the Pythagorean Theorem. If you are familiar with the length of the other two sides of the triangle that are called the legs, the theorem can be used to calculate the third side.

The formula to find out the length of Hypotenuse is the formal expression of the Pythagorean Theorem. It states that the total of the squares of the lengths of the two shorter sides of the right-angled triangle a and b is equivalent to the square of the length of the hypotenuse c:

a² + b² = c², where "a" and "b" are legs and "c" is the hypotenuse.

Calculate Hypotenuse from the Sides

You can observe from the hypotenuse formula for the Pythagorean theorem that taking the square root of each side provides a clear and precise formula for the value of the Hypotenuse

\[c=\sqrt{a^{2}+b^{2}}\]

Thus, if you already know the lengths of both legs of the triangle, you do not require having any detail about the value of the angles for the purpose of finding out the length of the Hypotenuse. All you have to do is square each leg value (the length) separately, add the results together, and take the square root of the obtained sum to get the answer.

Note: Do not make the mistake of summing up the values of both the legs first and then squaring the result or your answer will be incorrect.

Example: Find the length of the Hypotenuse of a right triangle of which the other two sides measures 8 cm and 15cm respectively

Solution: Given that the triangle is a right angled triangle

Calculate Hypotenuse from an Angle and a Side

The sides' equation of Hypotenuse is only of use unless you have the values of both legs. In certain circumstances, you may be given the length of only one leg in addition to the measure of 1 of the 2 non-right angles. This angle might be adjacent to the known side (leg), or it may be across from it

In an accurately labeled right triangle, side 'a' is placed between angle 'B' and the right angle 'C', and side 'b' is placed between angle 'A' and 'C'; the hypotenuse 'c', therefore links A and B. This contributes towards following trigonometric connection:

sin A = a/c, sin B = b/c

cos A = b/c, cos B = a/c

tan A = a/b, tan B = b/a

Fun Facts

• The term hypotenuse occurred from the Ancient Greek word hypoteinousa, which means 'extending under (a right angle)'. Fragment the term like this where, hypo- 'under', and teinein 'to extend'.

• The Hypotenuse will always be the longest side in a right triangle since it is opposite of the largest angle, the 90 degrees angle

• The shortest side in a right triangle is the one opposite to the smallest angle.

• You can follow a simple shortcut to find the shortest side. If you already know the shortest angle, then automatically the side opposite to it is the shortest.

• If you already know the medium angle, then the neighboring side to it is the shortest.

• It is mathematically NOT possible to solve a right triangle with only the Hypotenuse?

• Geometrical concepts are also quite useful in certain types of application in physics, engineering, navigation, construction etc. So, if you are aiming to be an engineer or interior designer, just do not take geometry for granted.

Conclusion:

The Hypotenuse is the longest side in a right triangle, opposite the largest angle. In certain cases, you may be given the length of only one leg in addition to the measure of 1 of the two non-right angles. This angle might be adjacent to the known side (leg) or across from it. By Pythagorean Theorem, the length of the Hypotenuse can be found by the equation h²=a²+b². When calculating the Hypotenuse, make sure to use the appropriate trigonometric function depending on which angle is given. Practising problems will help solidify your understanding of how to find the Hypotenuse.