5.8 SQUARE ROOTS & THE PYTHAGOREAN
THEOREM
Review:
Given a square that has a side whose length is 12 cm, find the
area.
12 cm
If we know that the area of a square is 25 sq. yds., what is the
length of each side?
25 sq yds
In this second case, we look for a number that when you
SQUARE it, you get 25. We say that number is the SQUARE
ROOT of 25.
The positive SQUARE ROOT of a positive number is a
POSITIVE number that when multiplied TIMES ITSELF
(squared), you will get that number.
Ex: The square root of 25 is 5 since 5 x 5 = 25
Using symbols, we write 25 = 5
You should know the square roots of all “perfect squares”
through 144 (lets generate that together, also on page A-2).
For right now, when we do NOT have a perfect square, we
will use a CALCULATOR to approximate the square root by
pressing the number, then the square root key on your
calculator, finally write your answer rounded to the nearest
thousandth.
Ex: approximate 27
When we press the correct keys on our calculator we get:
_______________ … (the periods show it continues, but we
don’t know how, the calculator has to stop somewhere),
which we would round to 5.__________ . (Nearest
thousandth.)
(See section 15.2 to find and write square roots without
using a calculator. THAT IS A VERY IMPORTANT
SECTION FOR ALGEBRA!)
Ex: Find the approximate square root of 125.
Ex: Simplify
9
64
(If time.) Now we will use this to apply it to finding
missing sides of a triangle using the PYTHAGOREAN
THEOREM. (Would you like to see a proof?)
The Pythagorean theorem states the relationship among
the LENGTHS of the SIDES of a RIGHT triangle as:
a2
+ b2
=
c2
a
c
b
Where a and b are the lengths of the sides, and c is the
length of the “hypotenuse” (side opposite the 90 degree
angle). The hypotenuse is ALWAYS LONGEST.
If we want to find the length of the hypotenuse given the
other two sides, we use:
hypotenuse =  (leg a)2 + (leg b)2
If we want to find the length of one of the legs given one
side and the hypotenuse we use:
leg =  (hypotenuse)2 - (other leg)2
Ex: Find the length of the missing side. (Remember you
are finding just the length of a side, so the units will be to
the _________power).
8 cm
?
6 cm
10 m
5m
?
Application: Lax Larry took the scenic route to his first
service call. From home he headed due south 30 miles,
then took a sharp left and headed east for 90 miles. On the
way home he took the street that went directly from the
service call to home. How far was he actually from work if
he takes the shortest route? (Draw a picture). (How many
extra miles did he bill his service call for?)
Add p 441 # 29 and 31 to your homework.