Square Roots and the Pythagoren Theorm

1.1 Square Numbers and Area Models

We can prove that 36 is a square number We can prove that 36 is a square number. Draw a square with an area of 36 square units. 6 units 36 = 6 x 6 = 62 62 = 36

We can prove that 49 is a square number We can prove that 49 is a square number. Draw a square with an area of 49 square units. 7 units 49 = 7 x 7 = 72 72 = 49

A square has an area of 64 cm2 Find the perimeter A square has an area of 64 cm2 Find the perimeter. 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 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 = 1 4 √4 = 2 9 √9 = 3 16 √16 = 4 25 √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 = 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? They all have an odd number of factors! They are perfect squares! 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 36 36 ÷ 2 = 18 2 and 18 are factors of 36 36 ÷ 3 = 12 3 and 12 are factors of 36 36 ÷ 4 = 9 4 and 9 are factors of 36 36 ÷ 6 = 6 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 Find the square root of 144. Find 42 List the factors of 121. Is there a square root? If so what is the square root? Which perfect squares have square roots between 1 and 50.

1.2 Quiz Find the square root of 144. 144 = 144/12 = 12 Find 42 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

Bingo, this one is closest!!!! 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 5.1 x 5.1 = 26.01 Therefore the √27 = approximately 5.2 Bingo, this one is closest!!!!

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 √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

Quiz 1.4 3 < √11 < 4 (√11 is closer to 3 than 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/geometryandmeasurement/pythagoreantheorem/preview.weml

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

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 + b2 = h2 a2 + b2 = h2 32 + 62 = h2 a2 + 42 = 82 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

Exploring the Pythagorean Theorem 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. a2 + b2 = h2 62 + 62 = 92 ? 36 + 36 = 81 ? ≠ 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 112 + 14.242 = 182 14.24 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 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. (4, 5, 6) a2 + b2 = h2 42 + 52 = 62 16 + 25 = 62 41 ≠ 36. No it is not a right triangle. (3, 4, 5) 32 + 42 = 252 9 + 16 = 25 25 = 25. Yes, it is a right triangle. (6, 8, 10) a2 + b2 = c2 62 + 82 = 102 36 + 64 = 100 100 = 100. Yes, it is a right triangle. List 2 Pythagorean triplets: ( 3 , 4 , 5 ) ( 5, 12, 13) ( 7, 24, 25) ( 8, 15, 17)

Applying the Pythagorean Theorm 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

Draw a diagram to solve Pythagorean Word Problems!! 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 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

a2 + b2 = h2 32 + b2 = 52 9+ b2 = 25 9+ b2 - 9 = 25 – 9 √b2 = √16 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 = 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 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? a2 + b2 = 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 + b2 = h2 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