Pythagorean theorem which states the special relationship between the sides of a right triangle is perhaps the most popular and most applied theorem in Geometry. The algebraic statement of the Pythagorean theorem is used to derive the distance formula in coordinate Geometry and to prove the Pythagorean identities in Trigonometry. In fact, the fundamentals of Trigonometry are taught using the ratios of the sides of a right triangle.

Right triangles and Pythagorean theorem are not only used to solve real life problems, but often used in solving many advanced problems in Mathematics and Physical Sciences.

The theorem was named after the Greek Mathematician Pythagoras who lived in 500 B.C in the year 1909. It is now known that Babylonians, Egyptians , Chinese and Indians knew the theorem earlier than the time of Pythagoras. Even though ancient sources agree that Pythagoras gave a proof for the theorem no original documents exist.
The statement of the theorem in proposition 47 of Euclid's Elements is as follows:
"In right angled triangles, the square on the side subtending the right angle is equal to the sum of the squares on the sides containing the right angle."
Euclid used squares drawn on the sides of the right angles and showed the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the legs of a right triangle.

Pythagorean History

The algebraic form of the statement of Pythagoras theorem c2 = a2 + b2 is used in solving right triangles.
The statement of the Pythagorean theorem is as follows:

In a right triangle the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

If Δ ABC is a right triangle,

Pythagorean Theorem Statement

then the hypotenuse is the side opposite to the right angle (AB) and the legs (BC and CA) are the sides containing the right angle.

Then according to Pythagorean theorem
BC2 + CA2 = AB2 or
a2 + b2 = c2.

The algebraic form of the Pythagorean theorem
c2 = a2 + b2.
is used as a formula to solve for the third side of a right triangle if the lengths of any two sides are given.
If the lengths of the legs are given, the formula used to find the length of the hypotenuse is
c = $\sqrt{a^{2}+b^{2}}$

When the lengths of hypotenuse and one of the legs is known, we use one of the following formula to solve for the second leg.
a = $\sqrt{c^{2}-b^{2}}$ or
b =$\sqrt{c^{2}-a^{2}}$

We will be using these formulas when solving few example problems on Pythagorean theorem.
The converse of Pythagorean theorem is also true and is stated as follows:
If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
Pythagorean Converse

If in triangle ABC , c2 = a2 + b2 where AB = c, BC = a and CA =b,
then triangle ABC is a right triangle right angled at the vertex C.

The Pythagorean theorem and its converse can be combined and stated as a single theorem as follows:
A triangle is a right triangle if and only if the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides.

Pythagorean Triple:
A Pythagorean triple is a set of three whole numbers a,b and c that satisfy the equation a2 + b2 = c2.

Examples:
(1) 3,4,5 (2) 7, 24, 25 (3) 5, 12, 13
If each number in a Pythagorean triple is multiplied by the same whole number, the resulting numbers also form a Pythagorean triple.

Thus (6, 8, 10), (300, 400, 500) and (10, 24, 26) are also Pythagorean triples.

A Pythagorean triple can represent the sides of a right triangle.

Conversely if the lengths of the sides of a right triangles are whole numbers, they form a Pythagorean triple.
Hundreds of Proofs exist for the Pythagorean theorem. Let us prove the theorem using the mean proportional property of the legs of a right triangle.

Given:
ABC is a right triangle right angled at vertex C.
c is the length of the hypotenuse AB and a and b
are the lengths of the legs, BC and CA.
Pythagorean Theorem Statement
Prove:
c2 = a2 + b2.

Pythagorean Proof
Statements Reason
1. ΔABC is a right triangle with
right angle at C.
1. Given

2. Draw CD AB.2. One and only one perpendicular can be drawn to a line from a point not lying on the line.
3. Let BD = x and AD = c-x
3. Segment addition Postulate.
4. $\frac{c}{a}$ = $\frac{a}{x}$
$\frac{c}{b}$ = $\frac{b}{c-x}$

4. In a right triangle the length of of a leg is the mean proportional of the lengths of the hypotenuse and the projection of that leg on the hypotenuse.
5. cx = a2 ---------(1) and
c(c - x) = b2 ⇒ c2 - cx = b2
---------(2)
5. Product of the means = Product of the extremes.
6. cx + c2 - cx = a2 + b2.
⇒ c2 = a2 + b2.
6. Addition Postulate.

The converse of Pythagorean theorem can be proved using a construction.
Given: ΔABC is a triangle with AB = c, BC = a and CA = b and
c2 = a2 + b2.
Prove: ΔABC is a right triangle right angled at C.

Pythagorean Converse Proof

Construction: Construct right triangle DEF with right angle at F and legs EF = a and FD = b.
DE2 = EF2 + FD2 = a2 + b2. applying Pythagoras theorem for right triangle DEF.
DE2 = c2 Substitution.
DE = c = AB
ΔABC ≅ ΔDEF SSS criterion of congruency.
∠C ≅ ∠F CPCTC.
Thus ∠C is a right angle angle ΔABC is a right triangle.
Finding the Length of the Hypotenuse:
Find the length x.
x is the length of the hypotenuse
corresponding to the value 'c' in the formula
c2 = a2 + b2.
c2 = 842 + 132.
= 7056 + 169
= 7225
c = √7225 = 85
Hence the length of the hypotenuse = 85 cms.
Pythagorean Example

Finding the length of a leg:
Pythagorean Example Problem
Find the length of the unknown leg in the adjoining diagram.
We can take a = x, b = 48 and c = 50.
a2 = c2 - b2.
x2 = 502 - 482.
= 2500 - 2304 = 196
x = √196 = 14
Hence the length of the leg = 14".

Find x in the following diagram rounded to the tenth of an unit.

Pythagorean Examples

The given triangle is an isosceles triangle. Hence the altitude bisects the base.
So we need to solve for one leg of a right triangle whose hypotenuse and the other leg measure 12 and 5 units respectively.
a2 = c2 - b2.
= 122 - 52.
= 144 - 25 = 119
a = √119 = 10.9 rounded to the tenth.

Determine whether the given set of numbers can be the lengths of the sides of a right triangle.
20, 21 and 29.
The greatest of the three numbers is 29 and 292 = 841.
202 + 212 = 400 + 441 = 841
Thus, 202 + 212 = 292.
Hence by the converse of Pythagorean theorem the three numbers can represent the lengths of the sides of a Right triangle,
Pythagorean theorem is used to find the lengths, distances and heights using right triangles which model real life situations.
When fire occurs in high raise buildings, the fire fighting men cannot use the regular stairs or lifts. They can reach some floors using ladders. In order to determine the ladder length, they apply the Pythagorean theorem as they can estimate the height of the floor affected and the horizontal distance they can use to keep the ladder in position.

Pythagorean Real Life Example

If you own a square plot and propose to construct a diagonal path along the path, the distance of the path can be found using Pythagorean theorem as the diagonal separates the square or a rectangle into two congruent right triangle and forms the hypotenuse for each of the triangle.

Pythagorean Real Life Examples
The distance between two points (x1, y1) and (x2, y2) d = $\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$
is derived using Pythagorean theorem on rectangular coordinate system.
Thus Pythagorean theorem is applied in any situation which fits a a right triangle model.

Solved Examples

Question 1: A ladder of length 16 ft is placed against a wall. If the foot of the ladder is 7 ft from the wall find the height the ladder reaches on the wall nearest to the tenth of a foot.

Pythagorean Word Problems
Solution:
 
The situation is described using the right triangle ABC, where AB represents the wall, BC the floor and AC the ladder.
    It is required to find the measure h, which is the length of a leg in right triangle ABC. Using Pythagorean theorem,

    h2 = AC2 - BC2.
         = 162 - 72 = 256 - 49 = 207.
    h = √207 = 14.4 ft ( rounded to the tenth of a foot).
 

Question 2: Roger is given a contract to lay tiles on a path which forms the diagonal of a square garden. If he is told that the length of the path required is 100√2 ft, what is the perimeter of the garden?

  Pythagorean Word Problems
Solution:
 
The diagonal of a Square separates it into two congruent right triangles and it forms the hypotenuse for both. The two adjacent sides of the square are thus the two congruent legs of the right triangle. Hence a = b = x and c = 100√2.
   Thus the Pythagorean formula for the situation is,
    x2 + x2 = (100√2)2.
    2x2 = 20000   ⇒ x2 = 10000  
     x = √10000 = 100 ft.
     Hence the side of the plot measure 100ft.
     Perimeter of the square garden = 4x = 4 x 100 = 400 ft.
 

Here are few practice problems to review your skills on Pythagorean theorem.

Practice Problems

Question 1: Find the value of x in the following triangles rounded to the tenth of a unit.

Pythagorean Practice Problems

  (Answers: 10.2 and 5.7)
Question 2: A ramp is placed from a ditch to a main road which is 2 ft above the ditch. If the length of the ramp is 12 ft how far away the bottom of the ramp from the Road?
   (Answer: 11.8ft)
Question 3: 20, 21, 29 is a Pythagorean triple. Using this check whether the numbers 60,63 and 87 can be the measures of the sides of a right triangle.
Question 4: A Satellite is orbiting the Earth at a height of 700 Kms. Assuming the radius of the earth to be 6370 Kms, find the distance of the Earth's Horizon from the Satellite nearest to a Km.

 Pythagorean Practice Problem

  (Answer: 3067 Kms).