How can the Pythagorean theorem be proved by paper folding?

Pythagorean theorem is a theorem for right-angled triangles, it is also referred to as the Pythagoras theorem. It is used to show the connection in the sides of a triangle which is a right-angled triangle. According to this theorem sum of squares of any two small sides is equal to the square of the biggest side. The small sides of a right-angled triangle are perpendicular and base while the biggest side is known as hypotenuse.

The discovery of this theorem is linked with an ancient Greek philosopher who was Pythagoras and hence it is called Pythagoras Theorem. 

Pythagoras Theorem Expression:

(Perpendicular)² + (Base)²  = (Hypotenuse)²

It can also be written in more general form as 

(side1)² + (side2)² = (side3)²

Here side1 and side2 is either perpendicular or base and side3 is the biggest side means hypotenuse.

Example: Sides of a right-angled triangle are given as  6, 8, 10, now check the Pythagorean theorem.

Solution:

Lets consider base = 6 and perpendicular = 8 so hypotenuse = 10

Sum of squares of base and perpendicular = 6² + 8² = 100

Square of the longest side = 10² = 100

So we can verify that

sum of squares of two small sides = square of the longest side 

(side1)² + (side2)² = (side3)²

                100 = 100

Proof By Paper Folding

Step 1: Take a square-shaped paper length of whose each side is (A + B). Cut out a small square of side C from it as shown in the below image.

Clearly, we cut out an area of C² so we got 4 right-angled triangles.   

Step 2: Now rearrange the 4 right-angled triangles, so move the triangle to the up as shown in the below figure. 

Step 3: Now move the below triangle to the left.

Step 4: Now move the upper triangle to down.

Step 5: Now see the below two figures in which C² is the sum of A² and B²

C² = A² + B²

Hence we proved Pythagoras Theorem.

Sample Questions

Question 1: Find the length of the hypotenuse of a right-angled triangle whose height is 4 cm and whose base is 3 cm?

Answer:  

Using Pythagorean theorem, a² + b² = c²

So 3² + 4² = c²

hence c = √(9 + 16)

         c = √25

         c = 5 cm

Length of hypotenuse = 5 cm

Question 2: Check if the given triangle is a right-angled triangle or not, sides are 11, 8, 6?

Answer:  

A right-angled triangle follows the Pythagorean theorem so use that.

Sum of squares of two small sides should be equal to the square of the longest side

So 8² + 6² must be equal to 11²

but 64 + 36 =100 while 11² = 121

Hence given triangle is not a right-angled triangle as it is not satisfying the Pythagorean theorem.

Question 3: Find the length of perpendicular of a right triangle whose hypotenuse is 29 cm and whose base is 20 cm?

Answer:

Applying Pythagoras theorem, a² + b² = c²

b(base)= 20, c(hypotenuse) = 29, find a(perpendicular)

so a = √(c² – b²)

hence a = √(29² – 20²)

         a = √841-400

         a = √441

         a = 21 cm

Length of perpendicular = 21 cm

Question 4: Check if the given triangle is a right-angled triangle or not, sides are 17, 8, 15?

Answer:  

A right-angled triangle follows the Pythagorean theorem so use that.

Sum of squares of two small sides should be equal to the square of the longest side

So 8² + 15² must be equal to 17²

but 64 + 225 =289 while 17² = 289

Hence given triangle is a right-angled triangle as it is satisfying the Pythagorean theorem.

Question 5: Find the length of the hypotenuse of a right-angled triangle whose height is 48 cm and whose base is 55 cm?

Answer:  

Using Pythagorean theorem, a² + b² = c²

So 48² + 55² = c²

hence c = √(2304 + 3025)

        c = √5329

        c = 73 cm

Length of hypotenuse = 73 cm