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The Pythagorean formula is actually a way to calculate the sides of a right triangle. So the problem you usually encounter is when we are asked to find the length of one of the sides of a right triangle.
History, Concept, Formula of the Pythagorean Theorem & Example Problems
The Pythagorean theorem formula is a fundamental relationship in geometry between the three sides of a right triangle. The Pythagorean Theorem is known to be the most popular relic of the mathematician Pythagoras.
However, there are pros and cons regarding the ownership and origins of the Pythagorean Theorem. Although in the end this theory was given to Pythagoras. Therefore, consider the history of the Pythagorean theorem formula below.
History of Pythagoras
Pythagoras (582 BC-496 BC) was born on the island of Samos, southern Greece. One of his most popular legacies is the Pythagorean theorem. Historically, 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 a discovery of Pythagoras because he was the first to prove the observation mathematically. Pythagoras also discovered another important thing in the field of mathematics, namely the golden ratio.
In the past, mathematics was not only concerned with numbers, but also to explain philosophy and understand 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 a human’s upper body compared to its lower body. Everything is close to the golden ratio of 1:1.618.
Mentioned in Tugino’s Penetrating the Wilderness of Pythagoras, Pythagoras proved that all objects that fulfill the golden ratio always have a very high level of aesthetics. The Pythagorean Theorem cannot be applied to all triangles.
This theorem applies to right triangles where the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other sides or right sides. In this way, the three sides of a right triangle have an interconnected relationship.
Pythagorean Theorem Concept
In a right triangle, the square of the hypotenuse is equal to the sum of squares on the other sides. But there is another theory, namely the opposite of the Pythagorean Theorem, which functions to determine the type of triangle if the length of the sides is known. So this type of triangle is:
Right triangles, which include triangles with one right angle of 90 degrees.
An acute triangle, all three acute angles are less than 90 degrees.
An obtuse triangle is a triangle whose angles are obtuse or measure more than 90 degrees.
Pythagorean Theorem Formula
The Pythagorean theorem formula relates to the elements of a right triangle. This element includes the right angled sides a and b as well as the length of the hypotenuse c. Thus it applies:
The square of the hypotenuse of a right triangle is equal to the sum of the squares of its two right sides. Or the length of the hypotenuse of a right triangle is equal to the square root of the sum of the squares of its right sides. So the formula is:
A2 + b2 = c2
Information:
C = hypotenuse
A = high
B = base
Pythagorean triples
Pythagorean triples are numbers that form a right triangle. This number also applies to multiples. Here are some Pythagorean triples:
3, 4, 5 and their multiples, (5 = hypotenuse)
5, 12, 13 and their multiples, (13 = hypotenuse)
8, 15, 17 and their multiples, (17 = hypotenuse)
7, 24, 25 and their multiples, (25 = hypotenuse)
20, 21, 29 and their multiples, (29 = hypotenuse)
9, 40, 41 and their multiples, (41 = hypotenuse)
11, 60, 61 and their multiples, (61 = hypotenuse
The Pythagorean formula is used to find out the value of the side opposite the right angle or hypotenuse. These two sides are also known as the hypotenuse.
In other words, it is important for you to know the basic concepts according to the laws mentioned previously. To apply the Pythagorean theorem, it can be used to determine the height of an equilateral triangle, determine the length of the diagonal of a square, rectangle, rhombus, diagonal of a block, cube, cone, etc.
Examples of Pythagorean Questions and Discussion
Some examples of Pythagorean theorem questions and discussions that students can learn when studying independently are as follows.
- A right triangle has a height of 9 cm with a base 12 cm long. Determine the slope of the right triangle.
Discussion:
A = 9 cm
B = 12 cm
C = ?
C2 = a2 + b2
C2 = 92 + 122
C2= 81 + 144
C= √255
C = 15
So the hypotenuse is 15 cm.
- There is a right triangle with two sides measuring 21 cm and 28 cm. Determine the length of the other side.
Discussion:
The side in question is the hypotenuse of a right triangle.
So that:
C2 = a2 + b2
= 212 + 282 = 441 + 784 = 1,225
C = √1.225 = 35 cm
The fast way:
By using triple (3, 4, 5) then each side of the triangle is multiplied by 7
(3 x 7, 4 x 7, 5x 7) so (21, 28, 35)
The length of the other side is 35 cm.
- The length of the hypotenuse of a right triangle is 2x + 2 cm. If the lengths of the other two sides are 4 cm and 2x + 1 cm, determine the value of x and the length of the hypotenuse.
Discussion:
To find the value of x, you can use the Pythagorean Theorem, namely:
BC2 = AC2 + AB2
(2x + 2)2 = 42 + (2x + 1)2
4×2 + 8x + 4 = 16 + 4×2 + 4x + 1
4×2 + 8x + 4 = 4 x 2 + 4x + 17
4x = 13
X = 13/4
X = 3.25
Then the satisfactory value of x is 3.25
The length of the hypotenuse is the length of BC, then:
BC = (2x + 2) cm
BC = (2 . 3.25 + 2) cm
BC = (6.5 + 2) cm
BC = 8.5 cm
So, the length of the hypotenuse is 8.5 cm
