6.2 The Pythagorean Theorems

One of the best known theorems in geometry (and all of mathematics for that matter) is the

Pythagorean Theorem. You have probably already worked with this theorem in your algebra

studies. The theorem is attributed to Pythagoras, a Greek mathematician and philospher, who

p

roved this theorem about twenty-five hundred years ago.

In order to understand the proof we must first investigate geometric means.

Proportions in which the means are equal occur frequently in geometry. For any two positive

numbers a and b, the geometric mean, of a and b is the positive number x such that

a

ÅÅ

Å

Å

x

=

x

ÅÅ

Å

Å

b

.

Example 1: Find the geometric mean of 4 and 8.

Solution:

4

Å

Å

Å

Å

x

=

x

Å

Å

Å

Å

8

Write the proportion

x

2

= 32 Use the Cross-Product Property

x =

è!!!!!!

32 Find the positive square root

x = 4

è!!!

2 Write in simplest radical form

The following theorem shows us how several geometric means "pop up" when you draw the

altitude to the hypotenuse of a right triangle.

Theorem: The altitude to the hypotenuse of a right triangle divides the triangle into two

triangles that are similar to the original triangle and to each other.

.

PythagoreanTheorems20052006.nb 1

Look at the figure below. We are given right DABC with CD

ê

êêê

ê

as the altitude to the hypotenuse.

A B

C

D

DABC ~ DACD ~ DCBD

This theorem can be easily proved by the AA Similarity Postulate since both smaller triangles

share an acute angle with DABC and all three triangles are right triangles. Since both smaller

triangles are similar to DABC, their corresponding angles are congruent. Thus the two smaller

triangles are similar to each other.

There are two important corollaries of this theorem one of which will be used in the proof of

the Pythagorean Theorem.

Corollary 1: When the altitude is drawn to the hypotenuse of a right triangle, the length of

the altitude is the geometric mean between the segments of the hypotenuse.

Corollary 2: When the altitude is drawn to the hypotenuse of a right triangle, each leg is

the geometric mean between the hypotenuse and the segment of the hypotenuse that is adja-

cent to the leg.

PythagoreanTheorems20052006.nb 2

Let's look at our triangle again.

A B

C

D

The first corollary states that CD is the geometric mean of AD and DB. Thus,

AD

ÅÅÅ

Å

Å

Å

Å

Å

Å

C

D

=

CD

ÅÅÅ

Å

Å

Å

Å

Å

Å

DB

.

The second corollary states that AC is the geometric mean of AB and AD (the segment of the

hypotenuse adjacent to AC) and that CB is the geometric mean of AB and DB. Thus,

AD

ÅÅÅ

Å

Å

Å

Å

Å

Å

A

C

=

AC

ÅÅÅ

Å

Å

Å

Å

Å

Å

AB

and

DB

ÅÅÅ

Å

Å

Å

Å

Å

Å

C

B

=

CB

Å

ÅÅ

Å

ÅÅ

Å

Å

Å

AB

N

ow we are ready to state and prove the Pythagorean Theorem.

Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to the

sum of the squares of the legs.

PythagoreanTheorems20052006.nb 3

Given: Right DABC; —C is a right —.

Prove: c

2

= a

2

+b

2

A B

C

ab

d e

c

Proof:

Statements Reasons

1. Draw a perpendicular from C to AB

êêêê

ê

1. Through a point outside a line, there is

exactly one line perp. to the line.

2.

c

Å

Å

Å

Å

a

=

a

Å

Å

Å

Å

e

;

c

Å

Å

Å

Å

b

=

b

Å

Å

Å

Å

d

2. Corollary 2 from above.

3. ce = a

2

; cd = b

2

3. A property of proportions

4. ce + cd = a

2

+b

2

4. Addition property

5. c(e + d) = a

2

+ b

2

5. Distributive property

6. c

2

= a

2

+b

2

6. Substitution property

Let's look at some examples.

PythagoreanTheorems20052006.nb 4

Example 2: Find the value of x.

3

6

x

Solution: x

2

= 6

2

+3

2

x

2

= 36 +9

x

2

= 45

x =

è!!!!!!

45

x = 3

è!!!

5 Note: we only used the positive square root

since x represents a length.

Example 3: Find the value of x.

x+2

x

10

Solution: x

2

+Hx +2L

2

= 10

2

x

2

+ x

2

+4

x +4 = 100

2

x

2

+4

x -96 = 0

x

2

+2

x -48 = 0

(x + 8)(x - 6) = 0

x = -8 or x = 6. Since x represents a length we discard the negative solution

and x = 6.

PythagoreanTheorems20052006.nb 5

The converse of the Pythagorean Theorem is also true and we present it, without proof, below.

The Converse of the Pythagorean Theorem: If the square of one side of a triangle is

equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Example 4: Determine if the triangle with side lengths 5, 12, and 13 is a right triangle.

Solution: 5

2

+12

2

= 25 +144 = 169 and 13

2

= 169 hence

5

2

+12

2

= 13

2

and by the converse of the Pythagorean Theorem the triangle is a right triangle.

Three integers (like 5,12, and 13) that satisfy the conditions of the Pythagorean Theorem are

called Pythagorean Triples. If the three integers are relatively prime (meaning they have no

common factors) then the three integers are know and Primitive Pythagorean Triples. Some

common Primitive Pythagorean Triples and their multiples are listed below. Memorizing

some of these triples can make your computations easier in the future.

Some Common Pythagorean Triples

3,4,5 5,12,13 8,15,17 7,24,25

6,8,10

9,12,15

12,16,20

10,24,26

When a triangle is not a right triangle, the squares of the sides can be used to determine

whether the triangle is obtuse or acute. Look at the theorems below.

Theorem: If the square of the longest side of a triangle is greater than the sums of the

squares of the other two sides, then the triangle is an obtuse triangle.

PythagoreanTheorems20052006.nb 6

A little common sense can tell us why this is true as well. Picture a right triangle.

a

b

c

The conclusion of the Pythagorean Theorem tells us that c

2

= a

2

+b

2

. Now picture the right

angle opening like a door without changing lengths a and b.

a

b

d

Obviously, the new longest side, d, is longer than the length of c in the first triangle. So, it

makes sense that d

2

> c

2

and thus d

2

> a

2

+b

2

.

Theorem: If the square of the longest side of a triangle is less than the sum of the squares

of the other two sides, then the triangle is an acute triangle.

N

ow picture the right angle closing.

a

b

e

PythagoreanTheorems20052006.nb 7

Obviously, in this case, e < c and thus e

2

< c

2

and e

2

< a

2

+b

2

. So we now have a way to

determine if a triangle is a right triangle, an obtuse triangle, or an acute triangle, if we know

the lengths of all three sides of the triangle.

Example 5: Determine whether a triangle formed with sides having the lengths named is

acute, right, or obtuse.

(A) 9, 40, 41 (B) 6, 7, 8 (C) 8, 10, 14

Solutions:

(A) 9

2

+40

2

?

41

2

81 + 1600

?

1681

1681 = 1681

The triangle is right.

(B) 6

2

+7

2

?

8

2

36 + 49

?

64

85 > 64

The triangle is acute.

(C) 8

2

+10

2

?

14

2

64 + 100

?

196

164 < 196

The triangle is obtuse.

N

ow we are ready to look at some special right triangles in the next section.

PythagoreanTheorems20052006.nb 8

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