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History, Concept, Formula of the Pythagorean Theorem & Example Problems
When you hear the term Pythagorean formula, what comes to your mind? Apart from a mathematical formula, surely you are curious about the history of this formula. Therefore, we have prepared it for you below.
History of the Pythagorean Theorem Formula
Pythagoras was born on the island of Samos, southern Greece. One of his most popular legacies is the Pythagorean theorem. Based on history, the contents of the Pythagorean theorem were actually known and applied by the Babylonians and Indians centuries before the birth of Pythagoras. However, the theorem is considered discovery of Pythagoras because he was the first to prove the observation a mathematically.
Pythagoras also discovered another important thing in the field of mathematics, namely the golden ratio or what is usually called the golden ratio. In ancient times, mathematics was not only related to numbers, but also to explaining philosophy and understanding beauty.
Apart from that, there are also objects that are said to lead to the golden ratio, for example snail shells, grooves on pineapples, and the size of human upper body compared to its lower body. Everything is close to the golden ratio of 1:1.618.
Pythagorean Theorem Concept
As explained, the Pythagorean theorem formula is a way to calculate the sides of a right triangle, where a right triangle has three sides, namely the base, the height, and the hypotenuse.
The Pythagorean theorem says:
“In right triangle, the square of the hypotenuse is a equal to the sum of the squares of the other sides.” In this way, the three sides of a right triangle have a mutually related relationship. This theorem has two mandatory properties, namely that it only applies to right triangles and you need to know at least two sides first to be able to determine the other side.
Meanwhile, the aim of this theory is basically to determine the length of one side of a right triangle. For a right triangle it is actually very easy because one of the characteristics of a right triangle is that one angle is 90o.
However, this theory can also be used to calculate the diagonal length of a square or cuboid and a cube that is shaped like a right triangle.
Pythagorean Theorem Formula
As explained above, the sound of the Pythagorean theorem states that for a right-angled triangle with markers a, b, c, then its sloped side is equal to the sum of the squares of the other sides.
For example, a triangle has base a and height b, then the slope is c. So, the sum of the squares of side c is equal to the sum of the squares of sides a and b. Based on the sound of the theory, here is how to calculate a right triangle using the Pythagorean theorem formula:
Find the slope:
C2 = a2 + b2
Finding the base side:
B2 = c2 – a2
Looking for high side or side:
A2 = c2 – b2
Or it could be interpreted that a is the high side, b is the base side, and c is the sloping side.
However, there are actually other ways to determine the sides of a right triangle. Namely by memorizing the Pythagorean triples. Because, this Pythagorean triple is the exact measurement of the sides of the Pythagorean triangle. The following numbers are called the Pythagorean triple:
3,4,5
5,12,13
6,8,10
7,24,25
8,15, 17
9, 12,15
10, 24, 26
Etc
So what is the meaning of this series of numbers? This means that if a right triangle has a height of 3 cm and a base of 4 cm, then you can be sure that the slope is 5 cm long.
By memorizing the series of Pythagorean triples, of course it will make it easier and shorten the time when we are working on a multiple choice question that does not require writing down the steps.
Example of a Pythagorean Theorem Question
To make it easier to learn about the Pythagorean formula, you should study the example questions directly. The more you study example questions and answers, the easier it will be for you to understand this theory.
First example: The hypotenuse of a right triangle is 16 units and one side of the triangle is 8 units. Find the size of the third side using this formula and explain it as best as possible so that it is easy to understand.
The solution is as follows:
Hypotenuse = 16 units
Perpendicular side height = 8 units
How to answer using the formula, namely
Hypotenuse 2 = base 2 + height 2
16 2 = b 2 + 82
B2 = 256 – 64
B = 192 = 13,856 units
It can be concluded that the length of the third side of a triangle is 13.856 units.
Second example: Maya wants to wash the windows of a building that is 12 feet off the ground. He also has a ladder that is 13 feet long. So how far should he put the base of the stairs from the building?
The solution to the Pythagorean formula is:
You can visualize the scenario to make it simpler and easier to understand. Try to find the base of the right triangle that has been formed and save it first.
The formula b2 = 132 – 122 knows that b is the distance of the base of the stairs from the foot of the building wall. So the conclusion is that b2 = 132 – 122 can be solved by b2 = 169 – 144 which is the same as 25 = 5. So it is known that the distance between the stairs and the building is 5 meters.
