Marcello Pedone The Pythagorean theorem

The Pythagorean theorem

Although Pythagoras is credited with the famous theorem, it is likely that the Babylonians knew the result for certain specific triangles at least a millennium earlier than Pythagoras. It is not known how the Greeks originally demonstrated the proof of the Pythagorean Theorem.

Marcello Pedone The Pythagorean theorem

The Pythagorean theorem

A B

C

hypotenuse

90°

Right triangle "In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs."

Marcello Pedone The Pythagorean theorem

The Pythagorean theorem

1 2Q Q Q

The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse.

2 2 2AB BC CA

Marcello Pedone The Pythagorean theorem

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

10

11

12

13 1

4 1

5 1

6

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

9(32)

16(42)

25(52)

2 2 25 3 4

25=9+16

Demonstrate the Pythagorean Theorem Many different proofs exist for this most fundamental of all geometric theorems

The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse.

B

C

AhypotenuseRight triangle

90°

Marcello Pedone The Pythagorean theorem

1

1

2

2

3

3

4

4

5

5

B

C

A

2 1 2Q 1 2 3 4 5Q

1 3 4 5Q

1 2Q Q Q The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse.

Several beautiful and intuitive proofs by shearing exist

Marcello Pedone The Pythagorean theorem

"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."

Marcello Pedone The Pythagorean theorem

The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse.

The Indian mathematician Bhaskara constructed a proof using the above figure, and another beautiful dissection proof is shown below

Marcello Pedone The Pythagorean theorem

Pythagorean Triples

Marcello Pedone The Pythagorean theorem

Pythagorean Triples There are certain sets of numbers that have a very special property.  Not only do these numbers satisfy the Pythagorean Theorem, but any multiples of these numbers also satisfy the Pythagorean Theorem.

For example:  the numbers 3, 4, and 5 satisfy the Pythagorean Theorem.  If you multiply all three numbers by 2  (6, 8, and 10), these new numbers ALSO satisfy the Pythagorean theorem.  

The special sets of numbers that possess this property are called Pythagorean Triples.

The most common Pythagorean Triples are: 3, 4, 5

5, 12, 13 8, 15, 17

Marcello Pedone The Pythagorean theorem

The formula that will generate all Pythagorean triples first appeared in Book X of Euclid's Elements:

where n and m are positive integers of opposite parity and m>n.

2 2

2 2

2x m n

y m n

z m n

Marcello Pedone The Pythagorean theorem

The Pythagorean theorem"In any right triangle, the square of the length of the hypotenuse is equal to the sum of the

squares of the lengths of the legs."

A triangle has sides 6, 7 and 10. Is it a right triangle?

The longest side MUST be the hypotenuse, so c = 10.  Now, check to see if the Pythagorean Theorem is true.

Since the Pythagorean Theorem is NOT true, this triangle is NOT a right triangle.

?2 2 2

?

10 6 7 ;

100 36 49

100 85

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The Distance Formula

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The distance between points P1 and P2 with coordinates (x1, y1) and (x2,y2) in a given coordinate system is given by the following distance formula:

2 2

1 2 1 2 1 2PP x x y y

1 2PP

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To see this, let Q be the point where the vertical line trough P2 intersects the

horizontal line trough P1.

• The x coordinate of Q is x2 , the same as that of P2.

• The y coordinate of Q is y1 , the same as that of P1.

• By the Pythagorean theorem .

2 2 2

1 2 1 2PP PQ PQ

Marcello Pedone The Pythagorean theorem

If H1 and H2 are the projection of P1 and P2 on the x axis, the segments P1Q and H1H2 are opposite sides of a rectangle ,

1 1 2PQ H H

But

so that

1 2 1 2H H x x so

1 1 2PQ x x Similarly,

2 1 2PQ y y

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2 2 2 2 2

1 2 1 2 1 2 1 2 1 2PP x x y y x x y y

Taking square roots, we obtain the distance formula:

2 2

1 2 1 2 1 2PP x x y y

1 1 2PQ x x

2 1 2PQ y y

Hence

2 2 2

1 2 1 2PP PQ PQ

Marcello Pedone The Pythagorean theorem

EXAMPLE

ABThe distance between points A(2,5) and B(5,9) is

2 2 2 25 2 9 5 3 4 9 16 25 5AB