Hypotenuse

Hypotenuse Meaning

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, A 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. 

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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. 

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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.

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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:

a^2 + b^2 = c^2a, where "a" and "b" are legs and "c" is the hypotenuse.

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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: The mathematical expression goes like below;-

                                                   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 resulting 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

Now, by Pythagorean Theorem,

(Side) ^2 + (side)^2 + (side) ^2 = (hypotenuse) ^2

Therefore, (8)^2 + (15) ^2 = 289

Thus, length of Hypotenuse = 17cms


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

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Quiz Time

Practice Problem

If the area of a right angled triangle is 40 sq.cm and its perimeter is 40 cm. Can you Find the length of its hypotenuse? 

Solution

Take the hypotenuse to be AC=h and the remaining two sides be AB=a and BC=b

Using Pythagorean Theorem, we have h2=a2+b2   

Given that: Area of △ABC=40sq.cm

Thus,21​×a×b=40

⇒ab=80   

Given that perimeter of △ABC=40 cm

That is a+b+h=40

⇒a+b=40–h

Squaring on both the sides, we obtain

(a+b)2=(40−h)2

⇒a2+b2+2ab=1600−80h+h2

⇒h2+2(80) =1600−80h+h2 

⇒160=1600−80h

⇒80h=1600–160=1440

Therefore, h=18 cm

Hence, the length of the square of a hypotenuse is 18 cm.


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.

FAQ (Frequently Asked Questions)

1. Can The Pythagorean Theorem Be Applied In Real Life?

Pythagorean theorem is also applied to find difficult to measure distances or lengths. For instance, if you know a grocery delivery truck had traveled 10 blocks north, then 6 blocks west to get from warehouse to the store, but you would like to know what the direct distance between warehouse and the store is. You could establish a mathematical expression of 10^2 + 6^2 = (the direct distance)^2 to identify that it is approximately12 blocks.

2. Is It Possible To Draw A Perpendicular To The Hypotenuse?

Yes, you can construct a perpendicular to the hypotenuse. Just get a compass, a ruler, and a pencil. Place the pair of your compasses to the hypotenuse (or any length, until it stays stable). Now, form a circle which has its median point at one of the ends of the hypotenuse. Draw another circle with its median point at the other end of the hypotenuse. Draw a line connecting the two points where these circles meet and you get a line. This line is your hypotenuse.