1
Professor Busken – Square Roots and the Pythagorean Theorem
Learning Objectives:
A Find the Square Root of a Number
B Approximate Square Roots
C Use the Pythagorean Theorem
Definition 1. The square of a number is the number times itself.
For instance, the square of 4 is 16 because 42 or 4 · 4 = 16. Another square of −4 is also
16 because (−4)2 = (−4) · (−4) = 16.
Definition 2. The reverse process of squaring is finding a square root.
For example, a square root of 16 is 4 because 42 = 16. Another square root of 16 is also
−4 because (−4)2 = (−4) · (−4) = 16.
Theorem 1. Every positive number has two square roots.
For instance, the square roots of 16 are 4 and −4.
√
, called a radical sign, to indicate the positive
Definition 3. We use the symbol
square root.
For example,
√
25 = 5 because 52 = 25 and 5 is positive.
√
9 = 3 because 32 = 9 and 3 is positive.
Theorem 2 (Square Root of a Number). The square root,
is the positive number b whose square is a. In symbols,
√
a = b if b2 = a
For example,
Find the square root.
√
1.
100
√
√
, of a positive number a
36 = 6 since 62 = 36
1.
2.
√
64
2.
3.
√
81
3.
2
Professor Busken – Square Roots and the Pythagorean Theorem
4.
√
5.
r
1
4
5.
6.
r
49
16
6.
7.
r
4
25
7.
8.
r
169
100
8.
121
4.
1 4
Definition 4. Numbers like , , 9 and 36 are called perfect squares because their
4 25
square root is a whole number or a fraction.
√
A square root such as 21 cannot be written as a whole number or a fraction since 21
is not a perfect square. It can be approximated by estimating, by using a table, or by
using a calculator.
√
21
√
0
0
√
1
√
1
4
2
√
9
3
√
16
4
√
25
5
√
36
6
√
49
7
√
64
8
Consider the diagram above. Above the number line, notice that as the numbers under
the radical signs increase their value, and thus their
√ placement on the number line,
increase too. We can use this fact to estimate that 21 is between whole numbers 4 and
5.
√
9. Without a calculator or table, approximate 51 to the nearest whole number.
9.
10.
Use a calculator to approximate
√
51 to the nearest one-thousandth.
10.
11.
Without a calculator or table, approximate
√
3 to the nearest whole number.
11.
12.
Use a calculator to approximate
√
3 to the nearest one-thousandth.
12.
3
Professor Busken – Square Roots and the Pythagorean Theorem
a2 + b2 = c2
Theorem 3. If a and b are the
lengths of the legs of a right triangle
and c is the length of the hypotenuse,
then a2 + b2 = c2 .
hypotenuse c
Leg a
(leg)2 + (leg)2 = (hypotenuse)2
Leg b
Without using a calculator, find the unknown length of each right triangle.
13.
leg = 6, hypotenuse = 10.
13.
14.
leg = 5, hypotenuse = 12.
14.
15.
leg = 9, hypotenuse = 12.
15.
Using a calculator, find the unknown length of each right triangle. Round
your answer to the nearest thousandth.
16.
leg = 30, hypotenuse = 72.
16.