Download 6.4:Square Roots and Pythagorean Theorem

6.4
Square Roots and the
Pythagorean
Theorem
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
The Square of a Number
The square of a number is the number
times itself.
The square of 6 is 36 because 62 = 36.
The square of –6 is also 36 because
(–6)2 = (–6) (–6) = 36.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
2
Square Root of a Number
The reverse process of squaring is finding a
square root.
A square root of 36 is 6 because 62 = 36.
A square root of 36 is also –6 because (–6)2 = 36.
We use the symbol
, called a radical sign, to
indicate the positive square root.
16  4 because 42 = 16 and 4 is positive.
25  5 because 52 = 25 and 5 is positive.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
3
Square Root of a Number
The square root,
, of a positive number a is the
positive number b whose square is a. In symbols,
a  b if b  a.
2
9  3 because 3  9.
2
Also, 0  0.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
4
Helpful Hint
Remember that the radical sign
is
used to indicate the positive square root
of a nonnegative number.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
5
Perfect Squares
1
4
Numbers like , 36,
, and 1 are
4
25
called perfect squares because their square
root is a whole number or a fraction.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
6
Approximating Square Roots
A square root such as 6 cannot be
written as a whole number or a fraction
since 6 is not a perfect square. It can be
approximated by estimating by using a
table or by using a calculator.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
7
Right Triangles
One important application of square roots has to do
with right triangles.
A right triangle is a triangle in which one of the angles is
a right angle or measures 90º (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.
hypotenuse
leg
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
leg
Martin-Gay, Prealgebra, 6ed
8
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
c
a
a b  c
2
2
2
b
In other words,
(leg)2 + (other leg)2 = (hypotenuse)2.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
9