Math 64
6.4 "Square Roots and the Pythagorean Theorem"
Objectives:
* Find the square root of a number.
* Approximate square roots.
* Use the Pythagorean Theorem.
Now that we know how to write ratios and solve proportions, in Section 6.5, we use proportions to help us …nd unknown
sides of similar triangles. In this section, we prepare for work on triangles by studying right triangles and their applications.
First, we need to review square roots.
Finding Square Roots
The square of a number is the number times itself. For example,
The reverse process of squaring is …nding a square root. For example,
Every positive number has two square roots. We see above that the square roots of 36 are 6 and 6: We use the
symbol
called a radical sign, to indicate the positive square root of a nonnegative number. For example,
;
:
Square Root of a Number:
The square root of a positive number a is the positive number b whose square is a.
p
In symbols,
Also, 0 = 0
:
Squares Roots of Perfect Squares From 1 to 400 :
p
p
p
1=
25 =
81 =
p
p
p
4=
36 =
100 =
p
p
p
9=
49 =
121 =
p
p
p
16 =
64 =
144 =
p
p
p
p
169 =
196 =
225 =
256 =
p
p
p
p
289 =
324 =
361 =
400 =
Example 1: (Finding square roots)
Findreach square root.
1
a)
4
b)
Page: 1
r
9
16
Notes by Bibiana Lopez
Prealgebra by Elayn Martin-Gay
6.4
Approximating Square Roots
1
9
; 121;
, and 1 are called perfect squares because their square root is a whole number or a fraction.
4 p 16
p
A square root such as 7 cannot be written as a whole number or a fraction since 7 is not a perfect square. Although 7
Numbers like
cannot be written as a whole number or a fraction, it can be approximated by estimating.
Example 2: (Approximating square roots)
Use a calculator to approximate each square root to the nearest thousandth.
p
p
a) 10
b) 62
Using the Pythagorean Theorem
One important application of square roots has to do with right triangles. Recall that a right triangle is a triangle in
which one of the angles is a right angle, or measures 90 (degrees). The hypotenuse of a right triangle is the side opposite
the right angle. The legs of a right triangle are the other two sides. The right angle in the triangle is indicated by the small
square drawn in that angle.
The Pythagorean Theorem:
If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then
2
2
In other words, (leg) + (other leg) = (hypotenuse)
2
Example 3: (Finding the hypotenuse of a right triangle)
Find the length of the hypotenuse of each of the following right triangles.
a)
b)
Page: 2
Notes by Bibiana Lopez
Prealgebra by Elayn Martin-Gay
6.4
Example 4: (Finding the missing leg of a right triangle)
Find the length of the missing leg of each of the following right triangles.
a)
b)
Example 5: (Application)
Find the height of the tree. Round the height to one decimal place.
Page: 3
Notes by Bibiana Lopez