Honors Geometry Section 5.4
The Pythagorean Theorem
In a right triangle the two sides
that form the right angle are called
the legs, while the side opposite
the right angle is called the
hypotenuse.
Consider placing four congruent right triangles
with legs a and b and hypotenuse c as shown at
the right. Notice that the large figure is a
square. Using the formula for the area of a
square (A = s2) what is its area?
A  (a  b)
2
A  ( a  b )( a  b )
A  a  ab  ab  b
2
A  a  2 ab  b
2
2
2
We can also find the area of the large
figure by adding the areas of the smaller
square and the four triangles. The area of
a triangle is found by the formula A  1 2 bh .
A smallersqu
are
c
2
A4 triangles  4 ( 1 ab )
2
A  c  2 ab
2
If we set the two expressions for the
area of the larger square equal to each
other, we get:
a  2 ab  b  c  2 ab
2
2
2
a b c
2
2
2
The Pythagorean Theorem
For any right triangle with hypotenuse
c and legs a and b, the sum of the
2
2
squares of the legs (a  b )is equal to
2
the square of the hypotenuse ( c ).
a b c
2
2
2
4 8  x
2
2
80  x
2
x
80
x
16
x4 5
5
2
A
1
bh
2
7  x  25
2
2
2
x  576
2
x
576  24
A
1
2
 24  7  84
A Pythagorean Triple is three
whole numbers that could be the
sides of a right triangle.
3 , 4 ,5
5 ,12 ,13
6 ,8 ,10 ,
10 , 24 , 26 , 14 , 48 ,50
9 ,12 ,15
15 , 36 , 39
7 , 24 , 25
21 , 72 , 75
Example: If a 25-foot ladder is leaning
against a house and the bottom of the
ladder is 9 feet away from the house, how
far up the side of the house is the top of
the ladder? Round to the nearest 1000th.
9  x  25
2
2
2
x  544
2
x
544  23 . 324
The converse of the Pythagorean Theorem is also true.
Pythagorean Theorem Converse
If the square of the largest side of a
triangle equals the sum of the
squares of the other two sides,
then the triangle is a right triangle.
If c  a  b , then  ABC is a right tria
2
2
2
ngle.
If a triangle is not a right triangle,
then it must be either acute or
obtuse.
If c  a  b , then  ABC is an obtuse triangle.
2
2
2
If c  a  b , then  ABC is an acute triangle.
2
2
2
Examples: Is a triangle with the given sides
acute, right, obtuse or can’t exist. If the triangle
cannot exist, explain why.
4 . 47
11.18
8.94
If longest side  sum of the other two sides, then the triangle cannot exist.
8
7 2
2
64
2

53
obtuse
2
5 5 
2
125
2 5   4 5 
2
20  80
125  100
obtuse
2
Examples: Is a triangle with the given sides
acute, right, obtuse or can’t exist. If the triangle
cannot exist, explain why.
7 . 07
5 2 
2
50
5 6
2
 61
acute
2
4  6  11
Does not exist
AC
2
8 3
AC
2
 73
2
AC 
2
73
73  AC  _____
11