Square Roots and the
Pythagoren Theorm
1.1
Square Numbers and Area Models
We can prove that 36 is a square number.
Draw a square with an area of 36 square units.
6 units
62 = 36
6 units
36 = 6 x 6 = 62
We can prove that 49 is a square number.
Draw a square with an area of 49 square units.
7 units
72 = 49
7 units
49 = 7 x 7 = 72
A square has an area of 64 cm2
Find the perimeter.
64 cm2
What number when multiplied by itself will give 64?
8 x 8 = 64
So the square has a side length of 8cm.
Perimeter is the distance around: 8 + 8 + 8 + 8 = 32
What is a Perfect Square? Part 1
Any rational number that is the square of another
rational number. In other words, the square root of
a perfect square is a whole number.
Perfect Square
Square Root
1
4
9
16
25
√1
√4
√9
√16
√25
=1
=2
=3
=4
=5
Perfect Squares
Use a calculator to determine if the following are
perfect squares
Perfect Square?
Square Root Per. Square?
√121
=
Y/N
√169
=
Y/N
√99
=
Y/N
√50
=
Y/N
Perfect Squares - KEY
Use a calculator to determine if the following are
perfect squares
Perfect Square?
Square Root Per. Square?
√121
=
11
Y/N
√169
=
13
Y/N
√99
=
9.95
Y/N
√50
=
7.07
Y/N
What is a Perfect Square? Part 2
Another way to look at it.
If we can find a division sentence for a number
so that the quotient is equal to the divisor, the
number is a square number.
16 ÷ 4 = 4
Dividend divisor quotient
Quiz #1 Ch 1
1) List the first 12 perfect squares.
2) If a square has a side length of 5cm, what is
the area? Show your work.
3) Find the side length of a square with an area
of 81 cm2. Show your work.
Quiz #1 Ch 1 Key
1) List the first 12 perfect squares.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
2) If a square has a side length of 5cm, what is the
area?
5cm x 5cm = 25cm2
3) Find the side length of a square with an area of
81 cm2.
81/9 = 9cm
1.2 Squares and Roots
Squaring and taking the square root are inverse
operations. That is they undo each other.
42 = 16
√16 = 16/4 = 4
Factors 1-30
What do you notice about all the yellow
columns?
1. They all have an odd number of factors!
2. They are perfect squares!
3. The middle factor is the square root of the
perfect square!
What is a perfect square? – PART 3
A perfect square will have its factor appear twice.
Ex:
36 ÷ 1 = 36
36 ÷ 2 = 18
36 ÷ 3 = 12
36 ÷ 4 = 9
36 ÷ 6 = 6
1 and 36 are factors of 36
2 and 18 are factors of 36
3 and 12 are factors of 36
4 and 9 are factors of 36
6 is a factor that occurs twice
Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36
The square root of 36 is 6 because it appears twice.
It is also the middle factor when they are listed in ascending
order!
What is a Perfect Square – Part 4
Is 136 a perfect square? Perfect squares have an odd number
of factors.
List the factors.
1 x 136 = 136
2 x 68 = 136
4 x 34 = 136
8 x 17 = 136
There are 8 factors in 136. Therefore, 136 is not a perfect
square because perfect squares have an odd number of
factors.
1.2 Quiz
1. Find the square root of 144.
2. Find 42
3. List the factors of 121. Is there a square
root? If so what is the square root?
4. Which perfect squares have square roots
between 1 and 50.
1.2 Quiz
1. Find the square root of 144.
2. Find 42
144 = 144/12 = 12
4x4=16
3.
List the factors of 121. Is it a PERFECT
SQUARE? If so what is the square root? Yes it is a
perfect square because there is an odd number of
factors. 1, 11 ,121. The square root is 11.
4.
What are the perfect squares between 1and 50.
1, 4, 9, 16, 25, 36, 49
1.3 Measuring Line Segments
1.3 Measuring Line Segments
1.3 Measuring Line Segments
1.3 Measuring Line Segments
1.3 Measuring Line Segments – Outside In
Practice Time
Complete the 7 questions below. You will be given a hard
copy (extra practice 1.3). You will need graph paper for
#4. Use inside-out for # 3 and outside-in for #4.
http://staff.argyll.epsb.ca/jreed/8math/unit1/03
.htm
1.4 Estimating Square Roots
1.4 Estimating Square Roots
Here is one way to estimate the value of the square root of a
number that is not a perfect square.
For example: Find √20
• Step 1: Is it a perfect square? No
• Step 2: If it isn’t, sandwich it between 2 perfect squares.
√16 < √20 < √25
4 < √20 < 5 (16, 17, 18, 19 20, 21, 22, 23, 24, 25: √20 is closer to 4 than 5)
Now we use guess and check.
• 4.6 x 4.6 = 21.16
• 4.5 x 4.5 = 20.25
• 4.4x 4.4 = 19.36
• Therefore the √20 = approximately 4.4/4.5
1.4 Estimating Square Roots
Here is one way to estimate the value of the square root of a
number that is not a perfect square.
For example: Find √20
• Step 1: Is it a perfect square? No
• Step 2: If it isn’t, sandwich it between 2 perfect squares.
√16 < √20 < √25
4 < √20 < 5 (16, 17, 18, 19 20, 21, 22, 23, 24, 25: √20 is closer to 4 than 5)
Now we use guess and check.
• 4.5 x 4.5 = 20.25 (too large)
• 4.4 x 4.4 = 19.36 (too small)
• 4.45 x 4.45 = 19.80
• 4.47 x 4.47 = 19.98
• Therefore the √20 = approximately 4.47
Another way to estimate √20
Find √27 (to one decimal place)
• Step 1: Is it a perfect square? No
• Step 2: If it isn’t, sandwich it between 2
perfect squares.
√25 < √27 < √36
5 < √27 < 6 - √27 is closer to 5 than 6
Now we use guess and check.
• 5.2 x 5.2 = 27.04
Bingo, this one is closest!!!!
• 5.1 x 5.1 = 26.01
Therefore the √27 = approximately 5.2
Find √105 (to 2 decimal places)
• Step 1: Is it a perfect square? No
• Step 2: If it isn’t, sandwich it between 2
perfect squares.
√100< √105 < √121
10 < √105 < 11 (√105 is closer to 10 than 11)
Now we use guess and check.
• 10.2 x 10.2 = 104.04 (not close enough 0.96)
• 10.3 x 10.3 = 106.09 (not close enough 1.09)
• 10.25 x 10.25 = 105.06
Therefore the √105 = approximately 10.25
Place each of the following square roots on the number line below.
√5, √52, and √89
√4< √5 < √9
2 < √5 < 3 - √5 is closer to 2 than 3
• 2.2 x 2.2 = 4.84
• 2.25x2.25 = 5.063
• 2.24 x 2.24 = 5.017
Bingo, this one is closest!!!!
• √5= approximately 2.24
√49 < √52 < √64 - √52 is closer to 7 than 8
• 7.2 x 7.2 = 51.8
• 7.25 x 7.25 = 52.56
• 7.22 x 7.22 = 52.12
• 7.21 x 7.21 = 51.98
Bingo, this one is closest!!!!
• √52= approximately 7.21
√81 < √89 < √100 - √83 is closer to 9 than 10
• 9.4 x 9.4 = 88.36
• 9.45 x 9.45 = 89.30
• 9.43 x 9.43 = 88.92 Bingo, this one is closest!!!!
• 9.44 x 9.44 = 89.12
• √89= approximately 9.43
√5
√89
Quiz 1.4
Estimate the square roots of √11 and √38
√9 < √11< √16
3 < √11 < 4 (√11 is closer to 3 than 4)
• 3.2 x 3.2 = 10.24
• 3.3 x 3.3 = 10.89
• 3.32 x 3.32 = 11.02
• √11= approximately 3.32
√36 < √38< √49
6 < √38 < 7 (√38 is closer to 6 than 7)
• 6.2 x 6.2 = 38.44
• 6.17 x 6.17 = 38.069
• 6.16 x 6.16 = 37.95
• √38= approximately 6.16
1.5 The Pythagorean Theorem
Watch Brainpop:
“Pythagorean Theorem”
http://www.brainpop.com/math/geometryand
measurement/pythagoreantheorem/preview.we
ml
1.5 The Pythagorean Theorem
In any right triangle, the area of the square
whose side is the hypotenuse is equal to the
sum of the areas of the squares whose sides are
the two legs
Watch This Video!
http://www.youtube.com/watch?v=0HYHG3fuzv
k
1.5 The Pythagorean Theorem
Side A and Side B are always the legs and they
are “attached” to the right angle. Side C is
always across from the right angle. It is always
longer than side A or side B. If you add the
squares of side A and B, it will = the square of
side C
1.5 The Pythagorean Theorem
The Formula(s)
a2 + b2 = c2 (most common)
c2 = a2 + b2
h2 = a2 + b2 (text book)
a2 + b2 = h2
1.5 The Pythagorean Theorem
Find the hypotenuse.
h2 = a2 + b2
h2 = 62 + 72
h2 = 36 + 49
h2 = 85
√h2 = √85
h = 9.22
1.5 The Pythagorean Theorem
Find the hypotenuse.
h2 = a2 + b2
h2 = 82 + 62
h2 = 64 + 36
h2 = 100
√ h2 = √100
h = 10
1.5 The Pythagorean Theorem
Find the leg “x”. We will make x – a.
h2 = a2 + b2
182 = a2 + 112
324 = a2 + 121
324 – 121 = a2 + 121 - 121
203 = a2
√203 = √a2
14.24 = a
1.5 The Pythagorean Theorem
Quiz
Find the missing sides
6 Steps
7 steps
Key
Find the missing sides
a2 + b 2 = h 2
a2 + b 2 = h 2
32 + 62 = h2
a2 + 42 = 8 2
9 + 36 = h2
a2 + 16 = 64
45 = h2
a2 + 16 – 16 = 64 - 16
√45 = √h2
a2 = 48
6.71 = h
√a2 = √48
a = 6.93
1.6
Exploring the
Pythagorean Theorem
For each triangle below, add up the 2 areas of the squares of the
legs in the 2nd column, and include the area of the square of the
hypotenuse in the third column. Do you see any patterns?
For each triangle below, add up the 2 areas of the squares of the
legs in the 2nd column, and include the area of the square of the
hypotenuse in the third column. Do you see any patterns?
Use Pythagoras to determine if the triangle
below is a right triangle.
a 2 + b2 = h 2
62 + 62 = 92 ?
36 + 36 = 81 ?
72≠ 81
This triangle is not a right triangle!
Use Pythagoras to determine if the
triangle below is a right triangle.
a2 + b2 = h2
72 + 242 = 252 ?
49 + 576 = 625 ?
625 = 625
This triangle is a right triangle! We can now say
that 7, 24, and 25 are Pythagorean triplets.
What is a Pythagorean Triplet?
It is a set of WHOLE numbers that satisfy the
Pythagorean theorem.
For example, this triangles’ sides (3, 4, 5) satisfy
the Pythagorean theorem and are therefore
triplets. This is because they are all whole
numbers and 32 + 42 = 52
What is a Pythagorean Triplet?
This triangles’ sides (6, 8, 11) do not satisfy the
Pythagorean theorem and are not therefore
triplets. Although they are all whole numbers,
they are not triplets because 62 + 82 ≠ 112
Pythagorean Triplets
This triangles’ sides are not Pythagorean triplets
because one of the sides is not a whole
number eventhough:
2
11
+
2
14.24
=
2
18
14.24
Your Turn!
In one minute, write down as many Pythagorean
triplets as you can where the hypotenuse is
less than 100.
Here are a few.
( 3 , 4 , 5 ) ( 5, 12, 13) ( 7, 24, 25) ( 8, 15, 17)
( 9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85)
(16, 63, 65) (16, 30 34) (20, 21, 29) (15, 20, 25)
(28, 45, 53) (33, 56, 65) (36, 77, 85) (39, 80, 89)
(48, 55, 73) (65, 72, 97)
Quiz 1.6
Are the triangles below right
triangles? The side lengths have
been given
A) 4, 5, 6
B) 3, 4, 5
C) 6, 8, 10
List 2 Pythagoren triplets:
Quiz 1.6
Are the triangles below a right triangle? The side lengths have been given. Show your
work.
1.
(4, 5, 6)
a2 + b2 = h2
42 + 5 2 = 6 2
16 + 25 = 62
41 ≠ 36. No it is not a right triangle.
2.
(3, 4, 5)
a2 + b2 = h2
32 + 42 = 252
9 + 16 = 25
25 = 25. Yes, it is a right triangle.
3.
(6, 8, 10)
a2 + b2 = c2
62 + 82 = 102
36 + 64 = 100
100 = 100. Yes, it is a right triangle.
1.
List 2 Pythagorean triplets:
1.7
Applying the
Pythagorean
Theorm
Find the missing side.
a2 + b2 = h2
42 + b2 = 72
16 + b2 = 49
16 + b2 – 16 = 49 – 16
b2 = 33
√b2 = √33
b = 5.74
Whenever Possible
Draw a diagram to solve
Pythagorean Word
Problems!!
Tanya runs diagonally across a rectangular field that has a
length of 40m and a width of 30m. What is the length
of the diagonal, in yards, that Tanya runs?
a2 + b2 = h2
302 + 402 = h2
900 + 1600 = h2
2500 = h2
√2500 = √h2
50 = h
To get from point A to point B you must avoid walking through a
pond. To avoid the pond, you must walk 34 meters south and
41 meters east. To the nearest meter, how many meters
would be saved if it were possible to walk through the pond?
a2 + b2 = h2
412 + 342 = h2
1156 + 1681 = h2
2837= h2
√2837= √h2
53.26 = h
Leo's dog house is shaped like a tent. The slanted sides
are both 5 feet long and the bottom of the house is 6
feet across. What is the height of his dog house, in
feet, at its tallest point?
a 2 + b 2 = h2
32 + b2 = 52
9+ b2 = 25
9+ b2 - 9 = 25 – 9
√b2 = √16
b=4
A ship sails 80 km due east and then 18 km due
north. How far is the ship from its starting
position when it completes this voyage?
802 + 182 = h2
6400 + 324 = h2
6724= h2
√6724 = √h2
82 = h
A ladder 7.25 m long stands on level ground so that the top
end of the ladder just reaches the top of a wall 5 m high.
How far is the foot of the ladder from the wall?
a 2 + b 2 = h2
a2 + 52 = 7.252
a2 + 25 = 52.56
a2 + 25 - 25 = 52.56 - 25
√a2 = √27.56
a = 5.25
1.7 Quiz
Draw a diagram and solve the following
problem. Approximate your answer to the
nearest tenths place: Two telephone poles are
75 m apart and the poles are each 20 m tall.
What is the distance from the base of one
pole to the top of the other pole?
1.7 Key
Draw a diagram and solve the following problem. Approximate your answer to the nearest tenths place:
Two telephone poles are 75 m apart and the poles are each 20 m tall. What is the distance from the
base of one pole to the top of the other pole?
a2 + b 2 = h 2
202 + 752 = h2
400 + 5625
6025= h2
77.6 = h
20 m
20 m
?
75m
1.7 Bonus
• If you were given just the side length of a
hypotenuse (√61cm), would you be able to
solve for leg a and leg b?
• The answers are a = 5cm, b=6cm