Transcript Square Roots and the Pythagoren Theorm

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?
25cm2
3) Find the side length of a square with an area of 81
cm2.
9cm
1.2 Squares and Roots
Squaring and taking the square root are inverse
operations. That is they undo each other.
42 = 16
√16 = √4x4 = 4
Factors 1-30
What do you notice about all the yellow columns?
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!
1.
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.
12
2.
Find 42
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 – Inside out.
1.3 Inside Out The Steps
 You can find the length of a line segment AB on a grid by constructing a square on
the segment. The length of AB is the square root of the area of the square.
Step 1 – Make a square around the line segment
Step 2 – Cut the square into 4 congruent triangles and a smaller square.
Step 3 – Calculate the area of the triangle A = bh/2 A = (3)(2)/2 A = 3 units
The area of one triangle is 3 units, so all triangles would be 4(3) = 12 units
Step 4 Calculate the are of a small square A = L x L = A = 1 x 1 = 1 unit
Step 5 Add the area of the squares and triangles together
12 + 1 = 13 so the line segment is the square root of 13
The Formula
A = l2 + 4 [(b)(h)/2]
1.3 Measuring Line Segments – Outside In
1.3 Outside In The Steps

You can find the length of a line segment AB on a grid by constructing a square on the segment.
The length of AB is the square root of the area of the square.
Step 1 – Make a square around the line segment
Step 2 – Draw a larger square around the line segment square.
Step 3 – Calculate the area of the outside square = l2 = 9 x 9 = 81
Step 4 – Calculate the area of the triangles (remember there are 4) 4 [(b)(h)/2] =
4 [(4)(5)/2] = 40
Step 5 Subtract the area of the triangles from the square.
81 – 40 = 41 so the line segment is the square root of 41
The Formula
A = l2 - 4 [(b)(h)/2]
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.
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 - √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
 4.47x 4.47 = 19.98
Bingo, this one is closest!!!!
 Therefore the √20 = approximately 4.47
Another way to estimate √20
Find √27
 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
 5.19 x 5.19 = 26.93
Bingo, this one is closest!!!!
Therefore the √27 = approximately 5.2
Find √105
 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
 10.25 x 10.25 = 105.06
 10.24 x 10.24 = 104.85
Bingo, this one is closest!!!!
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
 √52= approximately 7.21 Bingo, this one is closest!!!!
√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
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=0HYHG3fuzvk
Pythagorus
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
Some Questions
Find the hypotenuse.
a2 + b2 = c2
62 + 72 = c2
36 + 49 = c2
85 = c2
√85 = √c2
9.22 = c
We can now say that 6, 7, and 9.22 are not Pythagorean
triplets because one is not a whole number
Some Questions
Find the hypotenuse.
a2 + b2 = c2
82 + 62 = c2
64 + 36 = c2
100 = c2
√100 = √c2
10 = c
We can now say that 6, 8, 10 are Pythagorean triplets
Some Questions
Find the leg “x”. We will make x – a.
a2 + b2 = c2
a2 + 112 = 182
a2 + 121 = 324
a2 + 121 - 121= 324 - 121
a2 = 203
√a2 = √203
a = 14.24
We can now say that 11, 14.24, and 18 are not Pythagorean triplets,
because one is not a whole number.
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?
Use Pythagoras to determine if the
triangle below is a right triangle.
a 2 + b2 = c 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 = c2
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:
112 + 14.242 = 182
14.24
Your Turn!
In one minute, write down as many Pythagorean
triplets as you can where c (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)
1.7
Applying the
Pythagorean
Theorm
Find the missing side.
a2 + b2 = c2
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 = c2
302 + 402 = c2
900 + 1600 = c2
2500 = c2
√2500 = √c2
50 = c
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 = c2
412 + 342 = c2
1156 + 1681 = c2
2837= c2
√2837= √c2
53.26 = c
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?
a2 + b 2 = c 2
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 = c2
6400 + 324 = c2
6724= c2
√6724 = √c2
82 = c
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?
a2 + b 2 = c 2
a2 + 52 = 7.252
a2 + 25 = 56.56
a2 + 25 - 25 = 56.56 - 25
√a2 = √27.56
a = 5.25