Pythagoras Theorem

Definition

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle: the square of the hypotenuse is

equal to the sum of the squares of the other

two sides.

a

b

c

“C “ is the longest side of the

triangle

“a” and “b” are the two other sides

a2 +b2 =c2

Draw a DEF with l(DE) =3 cm, m E = 900,l(EF)= 4 cm Measure l(DF) it is 5cm.

3cm

4cm

5cm

D

E F

4cm

5cm

D

E F

Q

S

3 cm

Square DESQ of side 3 cm drawn on side DE. So the Area of DESQ = Side * Side = 3 * 3 = 9 Sq .cm

5cm

D

E F

Q

S

X Y

4 cm

3 cm

Square EFYX of side 4cm drawn on side EF. So the Area of EFYX = Side * Side , = 4 * 4 = 16 Sq .cm

5cm

D

E F

Q

S

X Y

4 cm

3 cm

Square DFMN = 5 * 5 = 25 sq.cm

M

N

DE

FQ

S

X Y 4 CM

3 CM

We add the area of square DESQ & EFYX We find that their sum is also 25 sq.cm.That is 32 + 42 = 52

E

Now you can use algebra to find any missing value, as in the following examples:

Example: Solve this triangle.a2 + b2 = c2

52 + 122 = c2

25 + 144 = c2

169 = c2

c2 = 169

c = √169

c = 13

Example: Solve this triangle.

a2 + b2 = c2

92 + b2 = 152

81 + b2 = 225

Take 81 from both sides:

b2 = 144

b = √144

b = 12

Example: What is the diagonal distance across a square of size 1?

a2 + b2 = c2

12 + 12 = c2

1 + 1 = c2

2 = c2

c2 = 2

c = √2 = 1.4142...

Pythagorean Theorem Algebra Proof

The Pythagorean Theorem states that, in a right triangle, the square of a (a2) plus the square of b (b2) is equal to the square of c (c2):

a2 + b2 = c2

We can show that a2 + b2 = c2 using AlgebraThe diagram has that "abc" triangle in it (four of them actually):

Area of Whole Square:It is a big square, with each side having a length of a+b, so the total area is:A = (a+b)(a+b)

Area of The Pieces :Now let's add up the areas of all the smaller pieces:

First, the smaller (tilted) square has an area of

  A = c2

    

And there are four triangles, each one has an

area of  A =½ab

So all four of them combined is

 A = 4(½ab) = 2ab

    

So, adding up the tilted square and the 4 triangles

gives:  A = c2+2ab

Both Areas Must Be Equal:The area of the large square is equal to the area of the tilted square and the 4 triangles. This can be written as:(a+b)(a+b) = c2+2abNOW, l rearrange this to see if we can get the pythagoras theorem:

Start with: (a+b)(a+b) = c2 + 2ab        

Expand (a+b)(a+b):

 a2 + 2ab +

b2= c2 + 2ab

        

Subtract "2ab" from both sides:

  a2 + b2 = c2

Therefore “In a right angled triangle , the area of the square on the Hypotenuse is equal to the sum of area of the square on the two remaining sides “

It works the other way around, too: when the three sides of a triangle make a2 + b2 = c2, then the triangle is right angled.

Example: Does this triangle have a Right Angle?

Does a2 + b2 = c2 ?a2 + b2 = 102 + 242 = 100 + 576 = 676c2 = 262 = 676They are equal, so ...Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle?Does 82 + 152 = 162 ?82 + 152 = 64 + 225 = 289, but 162 = 256So, NO, it does not have a Right Angle.

Therefore “In a right angled triangle , the area of the square on the Hypotenuse is equal to the sum of area of the square on the two remaining sides “