Name _____________________

Square Roots and the Pythagorean Theorem

Math 8Keywords

Square NumberPerfect Square

Square RootLeg

Hypotenuse Pythagorean Theorem

Pythagorean triple

What you’ll Learn

Determine the square of a number.

Determine the square root of a perfect square.

Determine the approximate square root of a non-perfect square.

Develop and apply the Pythagorean Theorem

Why It’s Important

The Pythagorean Theorem enables us to describe lengths that would be difficult to measure using a ruler. It enables a construction worker to

make a square corner without using a protractor.

Name _____________________

You will need to use graph paper for this unit.

Lesson 1.1 – Square Numbers and Area Models

Word Bank

Quadrilateral – any shape with _____________ and _______________.

Rectangle – a quadrilateral with _________________________.

Square – a quadrilateral with 4 right angles and _____________________.

Area - the amount of _________________________ an object. (example 25cm2)

Perimeter - the _________________________ an object. (example: 20cm)

Investigate

Using square tiles, make as many different rectangles as you can that are each 8 square units.

Draw the rectangles (with the units) below.

What are the lengths of each side of the rectangles?

Name _____________________

How is the length related to the area of the rectangle?

Connect

When we multiply a number by itself, we square the number.Example: The square of 4 is 4 x 4 = 16.So, 42 = 4 x 4 = 16We say: ______________________________________________16 is a square number, or a ______________________.

One way to model a square number is to draw a square whose area is equal to the square number.

Example: Show that 49 is a square number.

In order to find the perimeter of a square, we need to add all 4 sides together.

7 Units

7 Units

7 x 7 = _____

Or 72 = _____

We say: _____________________

Name _____________________

Example: A square picture has an area of 169cm2.Find the perimeter of the picture.

Step 1 - Find a number which when multiplied by itself, gives 169. This will be the length of one side.

Step 2 - Add all sides together to find the perimeter.

The perimeter of the picture is 52 cm.

Practice

We try You tryDraw as many different rectangles as you can with area 16cm2. Find the base and height of each rectangle.

Base Height Perimeter

Draw as many different rectangles as you can with area 20cm2. Find the base and height of each rectangle.

Base Height Perimeter

Using a diagram, show that 25 is a perfect square.

Using a diagram, show that 81 is a perfect square.

Name _____________________

Find the length of the diagram above. Find the length of the diagram above.

Assignment pg. 8-10 #1-6, 10-13, 19

Lesson 1.2 – Squares and Square Roots

Factor - A number that _____________________________ another number.

Example: 1, 2, 3, and 6 are factors of 6.What are the factors of 10?

Investigate

This chart shows the factors of each whole number from 1 to 8. Complete the chart.

6 8

4 3 4

Name _____________________

2 3 2 5 2 7 2

1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Which numbers only have 2 factors? What do you notice about these numbers?

Which numbers have an even number of factors, but more than 2 factors?

Which numbers have an odd number of factors? What do you notice about these numbers?

Connect

Ways to tell whether or not a number is a square number:

A. Find a division sentence so that the quotient (answer) is equal to the divisor (the number you are dividing the bigger number by).Example: 16 ÷ 4 = 4

B. Factoring. Factors of a number occur in pairsThese are the dimensions of a rectangle.Example: 16

Name _____________________

Sixteen has 5 factors: ___, ___, ___, ___, ___.Since there are an odd number of factors, one rectangle is square.The square has side length 4 units. We say that 4 is a square root of 16.We write 4 = √16

When a number has an odd number of factors, it is a _____________________.

When we multiply a number by itself, we _____________________ the number.

Squaring and taking the square root are ______________________ operations.They undo each other.Example: 4 x 4 = 16 √16=¿

so, 42 = ____ =

Practice

We try You tryFind the square root of 36. Find the square root of 49

1 and 16 are factors of 16

4 is a factor of 16. It occurs twice.

2 and 8 are factors of 16

Name _____________________

Find the square of 20. Find the square of 19.

List the factors of 169 in ascending order. Is 169 a square number?

List the factors of 196 in ascending order. Is 196 a square number?

Assignment Pg. 15-16 #3-8, 11-13, 19

Lesson 1.3 – Measuring Line Segments

Investigate

Name _____________________

a) Find the side lengths of each shape.

b) Find the area of each shape.

Connect

Area = Base =

Length = height =

Area of a square A = l 2

Side length of a square l¿√A

Area of a triangle A=

We can calculate the length of any line segment on a grid by thinking of it as the side length of a square.

There are 2 ways to draw the square:

1. By drawing INSIDE of the square 2. By drawing OUTSIDE of the square

Drawing INSIDE of the Square

Name _____________________

Given line segment PQ

Step 1 - use a ruler and a protractor to construct a perfect square on line segment PQ.The length of the line segment is the square root of the ________.

Step 2 - Cut the square into 4 congruent triangles and smaller squares.

Step 3 - The area of the triangle is A = b = and h = so, A =

= 4 units squared

Step 4 - Find the area of all 4 triangles. To do this, multiply the area of one triangle by 4.

Step 5 - Find the area of the small square.

Step 6 - Find the area of the triangles and the small square.Triangles = Small square =

Step 7 - Find the side length of the square.

Since ______ is not a square number, we cannot write ______ as a whole number.

Name _____________________

Drawing OUTSIDE of the Square

Given square ABCDa) Find the area of square ABCD. b) What is the length AB of the square?

Step 1 - Draw an enclosing square around square ABCD.

Step 2 - Find the area of square JKLM (the enclosing square)

Step 3 - Find the area one of the triangles formed by the enclosing

Step 4 - find the area of all 4 of the triangles formed by the enclosing square.

Step 5 - Find the area of square ABCD. To find the area, subtract the area of the triangles from the area of the enclosing square.

Step 6 - Find the side length AB

Name _____________________

Examples

We Try You TryGiven side length 6 cm, find the area of a square.

Given side length √7 cm, find the area of a square.

Given area 9 cm2 of a square, find the side length.

Given area 14 cm2 of a square, find the side length.

Find the area of square ABCD and then find the length of one side.

Find the area of square LMNO and then find the length of one side.

Name _____________________

Assignment Pg. 20-21 #3-7, 9-10Lesson 1.4 – Estimating Roots

Recall

Square Root √ Squared 2

Include in the

Answer

We include the √❑ symbol in the answer when the number that we are taking the square root of is ________________________________ and we want to leave the answer a whole number.

Example:

We include the 2 in our answer when speaking of area but only _____________________________. A 2 after units (cm, units, m, etc) indicates that we are speaking of an object that is two dimensional.

Example:

Note: 92 is not the same as 9cm2

Do NOT include in

the answer

We do not include the √❑ symbol in the answer when the number that we are taking the square root of ________________________________.

Example:

We do not include the 2 in our answer when speaking of ____________ because the length of something is one dimensional.

Example:

You will be marked incorrect if you write your answer using the wrong symbol.

Investigate

Using a number line, estimate where each square root may be placed.

√2 √5 √18 √24

Name _____________________

Connect

One strategy to estimate the value of the square root of a non-perfect square is to find the 2 perfect squares that the non-perfect square would be closest to (__________________)

Example: Estimate√20.

1. Find the square number that is closest to 20 but greater than 20.

2. Find the square number that is closest to 20 but less than 20.

3. On grid paper, draw both the greater than and the less than squares. Draw the squares so that they overlap.

4. A square with area 20 lies between these two squares.Its side length is¿¿.20 is between _____ and _____, but closer to _____.So, √20 is between ¿¿ and ¿¿ but closer to¿¿.So, √20 is between _____ and _____ but closer to _____.

5. An estimate of √20 is ____ to one decimal place. (Greater than ____, but closer to ____ than to ____).

Name _____________________

On a number line:

Practice

We Do You DoWhich whole number is √96 closer to? Which whole number is √72 closer to?

Estimate √40 Estimate √53

A square garden has area 139 m2.What are the approximate dimensions of the garden to 2 decimal places?

A square yard has area 111 m2.What are the approximate dimensions of the yard to 2 decimal places?

Name _____________________

Calculate how much fence you would need to go around the garden.

Calculate how much fence you would need to go around the yard.

Assignment - pg. 25 - 26 # 4, 5, 8-14

Lesson 1.5 – The Pythagorean Theorem

Isosceles Right Triangle Scalene Right Triangle

A right triangle has 2 sides that make a right angle. The other side is called a ________________. The two shorter sides are called _______.

You can use the properties of a right triangle to find the length of a line segment.

Connect

Here is a right triangle with a square drawn on each side.

Name _____________________

Note that ___________A similar relationship is true for all right triangles.

In a right triangle, the area of the square on the hypotenuse is equal to the sum of the area of the squares on the legs.

This relationship is called the Pythagorean Theorem.Formula is a2 + b2 = c2 where a and b are the _______ and c is the ______________.

We can use this relationship to find the length of any side of a right triangle, when we know the lengths of the other two sides.

Example

Find the length of the hypotenuse.

1. Label the hypotenuse c.The area of the square on the hypotenuse is c2.

2. Find the area of the squares on the legs.

3. The area of the hypotenuse square is the area of the squares of the legs added together.

4. The length of the hypotenuse is the square root of the hypotenuse. Use a calculator to find the root.

Practice

Name _____________________

We Do You Do

Find the length of the hypotenuse. Find the length of the hypotenuse.

We do You doFind the length of the hypotenuse. Find the length of the hypotenuse.

Name _____________________

Find the length of the leg. Find the length of the leg.

Assignment - pg. 34-35

Lesson 1.6 – Exploring the Pythagorean Theorem

Name _____________________

Connect

For each triangle, compare the sum of the areas of the 2 smaller squares to the area of the largest square.

Notice that the Pythagorean Theorem is only true for the ___________ triangle.

We can use this information to figure out if a triangle is a right triangle or not.

Triangle

Sum of areas of the

2 smaller squares

Area of the larger square

Acute

Right

Obtuse

Name _____________________

If the area of A + the area of B = the area of C, thenthe triangle is a right triangle.or if a2 + b2 = c2 then it’s a right triangle.

If the area of A + the area of B ≠ the area of C, thenthe triangle is not a right triangle.or if a2 + b2 ≠ c2 then it’s not a right triangle.

o the ≠ means does not equal

Steps

To determine whether a triangle with given side lengthsis a right triangle, we:

1. Sketch the triangle with a square on each side. The square with the longest side is square C.

2. Find the area of each square.

3. Add squares A and B together.

4. If A + B = C, then the triangle is a right triangle.

When a set of 3 numbers satisfies the Pythagorean Theorem, it is called a _______________________.

Practice

Square B

Square A

Square C

Name _____________________

We Do You DoDetermine if the following is a right triangle. Determine if the following is a right triangle.

Determine whether the triangle with sides 6cm, 6cm, 9cm is a right triangle.

Determine whether the triangle with sides 7cm, 24cm, 25cm is a right triangle

Assignment – pg. 43-44 #

Name _____________________

Lesson 1.7 – Applying the Pythagorean Theorem

Investigate

A doorway is 2.0 m high and 1.0 m wide. A square piece of plywood has side length 2.2 m. Can the plywood fit through the door?How do you know?Show your work.

Draw a picture of the problem Solve the problem using numbers Sentence answer

Helpful tips when solving story problems involving triangles,

Read the question carefully. Draw a picture of the story problem and label all given sides. Use the information in your picture to solve using the Pythagorean

Theorem, Write a sentence to explain your answer.

5 cm

a2 + b2 = c2

The length of leg a is ____ cm.a

Name _____________________

Example 1 – We Do

Marian helped her dad build a small rectangular table for her bedroom. The tabletop has length 56 cm and width 33 cm. The diagonal of the tabletop measures 60 cm.Does the tabletop have square corners?How do you know?

Draw a picture of the problem Solve the problem using numbers Sentence answer

Example 2 – You Do

A ramp is used to load a snow machine onto a trailer. The ramp has horizontal length 168 cm and sloping length 175 cm.The side view is a right triangle.How his is the ramp?

Draw a picture of the problem Solve the problem using numbers Sentence answer

4 cm

Name _____________________

Assignment – pg. 49-50 # 5 – 9, 11, 13, 16