THE PYTHAGOREAN THEOREM

THE PYTHAGOREAN THEOREM

Pythagorean theorem

The well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a² + b² = c². Although the theorem has long been associated with Greek mathematician-philosopher Pythagoras (c. 570–500/490 BCE), it is actually far older. Four Babylonian tablets from circa 1900–1600 BCE indicate some knowledge of the theorem, with a very accurate calculation of the square root of 2 (the length of the hypotenuse of a right triangle with the length of both legs equal to 1) and lists of special integers known as Pythagorean triples that satisfy it (e.g., 3, 4, and 5; 3² + 4² = 5², 9 + 16 = 25). The theorem is mentioned in the Baudhayana Sulba-sutra of India, which was written between 800 and 400 BCE. Nevertheless, the theorem came to be credited to Pythagoras. It is also proposition number 47 from Book I of Euclid’s Elements.

According to the Syrian historian Iamblichus (c. 250–330 CE), Pythagoras was introduced to mathematics by Thales of Miletus and his pupil Anaximander. In any case, it is known that Pythagoras traveled to Egypt about 535 BCE to further his study, was captured during an invasion in 525 BCE by Cambyses II of Persia and taken to Babylon, and may possibly have visited India before returning to the Mediterranean. Pythagoras soon settled in Croton (now Crotone, Italy) and set up a school, or in modern terms a monastery (see Pythagoreanism), where all members took strict vows of secrecy, and all new mathematical results for several centuries were attributed to his name. Thus, not only is the first proof of the theorem not known, there is also some doubt that Pythagoras himself actually proved the theorem that bears his name. Some scholars suggest that the first proof was the one shown in the figure. It was probably independently discovered in several different cultures.

Proof of Pythagorean Theorem

The proof of Pythagorean Theorem in mathematics is very important.

In a right angle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

States that in a right triangle that, the square of a (a²) plus the square of b (b²) is equal to the square of c (c²).

In short it is written as: a² + b² = c²

Proof of Pythagorean Theorem using Algebra:

Given: A ∆ XYZ in which ∠XYZ = 90°.

To prove: XZ² = XY² + YZ²Construction: Draw YO ⊥ XZ

Proof: In ∆XOY and ∆XYZ, we have,

∠X = ∠X              → common

∠XOY = ∠XYZ          → each equal to 90°

Therefore, ∆ XOY ~ ∆ XYZ → by AA-similarity

⇒ XO/XY = XY/XZ               

⇒ XO × XZ = XY² ----------------- (i)

In ∆YOZ and ∆XYZ, we have,

∠Z = ∠Z                  →           common

∠YOZ = ∠XYZ              →           each equal to 90°

Therefore, ∆ YOZ ~ ∆ XYZ          →          by AA-similarity

⇒ OZ/YZ = YZ/XZ                 

⇒ OZ × XZ = YZ² ----------------- (ii)

From (i) and (ii) we get,

XO × XZ + OZ × XZ = (XY² + YZ²)

⇒ (XO + OZ) × XZ = (XY² + YZ²)

⇒ XZ × XZ = (XY² + YZ²)

⇒ XZ ² = (XY² + YZ²)

Applying the Pythagorean theorem

Example:
Find the value of 
x.

Solution:
The side opposite the right angle is the side labelled x
x. This is the hypotenuse. When applying the Pythagorean theorem, this squared is equal to the sum of the other two sides squared. Mathematically, this means:

62 + 82 = x2

Which is the same as:
100=x2

Therefore, we can write:
x =  $\sqrt{100}$  
x = 10

Example:
Find the value of y.

Solution:
The side opposite the right angle has a length of 12.

Therefore, we will write:
82 + y2 = 122

This is the same as:
64+y2 =144

Subtracting 64 from both sides:
y2 =80

Therefore:
y = $\sqrt{80}$  
   =  $\sqrt{16\ \times\ 5}$  

   =  $4\sqrt{5}$  
In this last example, we left the answer in exact form instead of finding a decimal approximation. This is common unless you are working on an applied problem.

Applications (word problems) with the Pythagorean theorem

There are many different kinds of real-life problems that can be solved using the Pythagorean theorem. The easiest way to see that you should be applying this theorem is by drawing a picture of whatever situation is described.

Example:
Two hikers leave a cabin at the same time, one heading due south and the other headed due west. After one hour, the hiker walking south has covered 2.8 miles and the hiker walking west has covered 3.1 miles. At that moment, what is the shortest distance between the two hikers?

Solution:
First, sketch a picture of the information given. Label any unknown value with a variable name, like x.

Due south and due west form a right angle, and the shortest distance between any two points is a straight line.

Therefore, we can apply the Pythagorean theorem and write:
3.12 + 2.82 = x2

Here, you will need to use a calculator to simplify the left-hand side:

17.45=x2

Now use your calculator to take the square root. You will likely need to round your answer.

x =  $\sqrt{17.45}$  

 $\approx$  4.18 miles