The Pythagorean
Theorem
A tool for right
triangle problems
only
When do I need to use
the Pythagorean
Theorem ?
When I know: length of 2 sides
and
Need to know: length of 3rd side
Pythagorean Theorem
In a right triangle, the square of the length
of the hypotenuse is equal o the sum of
the squares of the lengths of the legs.
What is the Pythagorean
Theorem?
a2 b2 c2
Where a and b are legs and c is the hypotenuse
leg2 + leg2 = hyp2
Sides are always positive numbers
Practice with Radicals Work with the Pythagorean Theorem often requires us
to work with radicals.
Simplify each expression:
A) (√3) 2 √3 × √3
√9 3
B) ( 3 √ 11 ) 2 3 √ 11 × 3 √ 11 9 × √ 121 9 × 11 99
Your Turn
Simplify each expression:
A. (√ 5) 2
B. (2 √ 7) 2
C. (7 √ 2 ) 2
D. (2n) 2
E. 2
5
3
F.
G. 2
63
2
2
2
2
The Answers
Simplify each expression:
A. (√ 5) 2
B. (2 √ 7) 2
C. (7 √ 2 ) 2
D. (2n) 2
E. 2
5
3
F.
G. 2
63
2
2
2
2
= 5
= 28
= 98
= 4n2
= 9/5
= 1/2
= 24/9
Pythagorean Triples In every triple, the largest # is
the length of the hypotenuse
and the 2 smaller numbers are
the lengths of the legs of the
right triangle.
Using a2 + b2 = c2
Looking for length of
the hypotenuse a2 + b2 = c2
152 + 202 = c2
225 + 400 = c2
625 = c2
625 c2
25 c
Using a2 + b2 = c2
Looking for length of
A leg
a2 + b2 = c2
62 + b2 = 102
36 + b2 = 100
-36 -36
b2 = 64
b = 8
The Converse of Theorem If the square of 1 side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
c
b
a
More simply…
if c2 = a2 + b2,
then the triangle is a
right triangle.
Pythagorean Inequalities Theorems If c2 > a2 + b2,
then the triangle is obtuse.
If c2 < a2 + b2,
then the triangle is acute.
Applying the
Pythagorean Theorem
Practice #1. A 15 foot ladder leans up
against a building. The foot of the
ladder is 5 feet from the base of the
building. How high up the wall, to the
nearest foot does the ladder reach?
Draw a picture:
Solving the problem
a2 + b2 = c2 Write formula
x2 + 52 = 152 Substitute in
x2 + 25 = 225 Solve for x
- 25 -25
x2 = 200
x = 14.142135
x 200
The ladder reaches 14 feet up the wall.
Practice Problem #2.
Tim rode 8 miles due north, then 3 miles
due east. How far, to the nearest mile, is
Tim from where he started?
Draw a picture:
Solve the problem
a2 + b2 = c2
82 + 32 = c2
64 + 9 = c2
73 = c2
73 c
C = 8.5440037
Practice Problem #3.
As seen in the
accompanying diagram, a
person can travel from
NYC to Buffalo by going
north 170 miles to Albany
and then west 280 miles to
Buffalo.
A) If a highway is built to
connect NYC and Buffalo,
how many miles would be
saved on the trip?
Albany
New York City
170 miles
280 miles
???
Buffalo
Find length of new highway
Buf Albany
New York City
170 miles
280 miles
???
a2 + b2 = c2
2802+1702=c2
107300 m= c2
107300 c2
327.566= c
Did I answer question?
How many miles would be saved?
Old Distance: 280 + 170 = 450
New Distance: 327.566
Saved Miles: 122.4 or 122 miles