square roots and the pythagoren theorm. 1.1 square numbers and area models

Square Roots and the Pythagoren Theorm

1.1Square 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

6 units36 = 6 x 6 = 62

62 = 36

We can prove that 49 is a square number.

Draw a square with an area of 49 square units.

7 units

7 units49 = 7 x 7 = 72

72 = 49

A square has an area of 64 cm2

Find the perimeter.

What number when multiplied by itself will give 64?8 x 8 = 64So the square has a side length of 8cm. Perimeter is the distance around: 8 + 8 + 8 + 8 = 32

64 cm2

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 √1 = 14 √4 = 29 √9 = 316 √16 = 425 √25 = 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 = 4Dividend 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?

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 1 and 36 are factors of 3636 ÷ 2 = 18 2 and 18 are factors of 3636 ÷ 3 = 12 3 and 12 are factors of 3636 ÷ 4 = 9 4 and 9 are factors of 3636 ÷ 6 = 6 6 is a factor that occurs twice

Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36The 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 = 1362 x 68 = 1364 x 34 = 1368 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 segmentStep 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 unitsThe area of one triangle is 3 units, so all triangles would be 4(3) = 12 unitsStep 4 Calculate the are of a small square A = L x L = A = 1 x 1 = 1 unitStep 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 segmentStep 2 – Draw a larger square around the line segment square.Step 3 – Calculate the area of the outside square = l2 = 9 x 9 = 81Step 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 √20Step 1: Is it a perfect square? NoStep 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 = 21.164.5 x 4.5 = 20.254.47x 4.47 = 19.98 Therefore the √20 = approximately 4.47

Bingo, this one is closest!!!!

Another way to estimate √20

Find √27Step 1: Is it a perfect square? NoStep 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 = 27.045.19 x 5.19 = 26.93 Therefore the √27 = approximately 5.2 Bingo, this one is closest!!!!

Find √105Step 1: Is it a perfect square? NoStep 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 = 104.0410.25 x 10.25 = 105.0610.24 x 10.24 = 104.85 Therefore the √105 = approximately 10.25 Bingo, this one is

closest!!!!

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 √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√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 - 9.44 x 9.44 = 89.12 √89= approximately 9.43

Bingo, this one is closest!!!!

Bingo, this one is closest!!!!

Bingo, this one is closest!!!!

√5 √52 √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 = cWe 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.

a2 + b2 = c2

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 = 4916 + b2 – 16 = 49 – 16b2 = 33√b2 = √33b = 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 + b2 = c2

32 + b2 = 52

9+ b2 = 259+ b2 - 9 = 25 – 9√b2 = √16b = 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 + b2 = c2

a2 + 52 = 7.252

a2 + 25 = 56.56a2 + 25 - 25 = 56.56 - 25 √a2 = √27.56a = 5.25