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Pythagoras theorem states that for all right-angled triangles, 'The square on the hypotenuse is equal to the sum of the squares on the other two sides'. The hypotenuse is the longest side and it's always opposite the right angle.

1.
Pythagorean Theorem

MCC8.G.6-8: Apply the Pythagorean Theorem

to determine unknown side lengths in right

triangles in real-world and mathematical

problems in two and three dimensions.

MCC8.G.6-8: Apply the Pythagorean Theorem

to determine unknown side lengths in right

triangles in real-world and mathematical

problems in two and three dimensions.

2.
No need

For notes

Essential ?? On this slide

• How can we use the

Pythagorean Theorem to

solve for a missing length

of a right triangle.

For notes

Essential ?? On this slide

• How can we use the

Pythagorean Theorem to

solve for a missing length

of a right triangle.

3.
Warm - Up

Solve for x

• x2+7=43

• 64+x2=164

Evaluate for a = 12, b = 5, c = 13

3. a2 + b2

4. c2 – b2

Solve for x

• x2+7=43

• 64+x2=164

Evaluate for a = 12, b = 5, c = 13

3. a2 + b2

4. c2 – b2

4.
Here we have a triangle with

the lengths of each of the

three sides

5

4

3

the lengths of each of the

three sides

5

4

3

5.
Let’s take the lengths

of each side and make

a square for each of

them

5

4

3

of each side and make

a square for each of

them

5

4

3

6.
5

Let’s find the area of 4

10

each square? 3

9 15

2 8

1 14 20

1 2 3 4 7 13

6 19 25

12 18

5 6 7 8 24

11 17 23

9 10 11 12 16 22

13 14 15 16 21

1 2 3

4 5 6

7 8 9

Let’s find the area of 4

10

each square? 3

9 15

2 8

1 14 20

1 2 3 4 7 13

6 19 25

12 18

5 6 7 8 24

11 17 23

9 10 11 12 16 22

13 14 15 16 21

1 2 3

4 5 6

7 8 9

7.
Now, let’s add the two

smaller areas

together.

25

16

+ 9

smaller areas

together.

25

16

+ 9

8.
Notice how the sum of

the two smaller squares

equals the larger

square?

25

It turns out

9+16 = ?

this is true

for every

right triangle

the two smaller squares

equals the larger

square?

25

It turns out

9+16 = ?

this is true

for every

right triangle

9.
The Pythagorean Theorem states: “The

sum of the squares of the legs of a right

triangle are equal to the square of the

hypotenuse.”

9+16 = 25

sum of the squares of the legs of a right

triangle are equal to the square of the

hypotenuse.”

9+16 = 25

10.
No need

For notes

Pythagorean Theorem On this slide

• What is the Pythagorean Theorem in

symbol form?

a +b =c

2 2 2

• Which of these variables represent the

hypotenuse?

c

• Once you have figured out which is c,

does it matter which leg is a and which

is b?

no

For notes

Pythagorean Theorem On this slide

• What is the Pythagorean Theorem in

symbol form?

a +b =c

2 2 2

• Which of these variables represent the

hypotenuse?

c

• Once you have figured out which is c,

does it matter which leg is a and which

is b?

no

11.
TAKE

Steps to Solve for a missing side NOTES

of a right triangle using the

Pythagorean Theorem

The following are the basic steps for solving a

Pythagorean Theorem Problem.

Step 1: Write the formula

Step 2: Substitute known values for the

variables.

Step 3: Solve for the missing variable.

Lets break this down a little further…

Steps to Solve for a missing side NOTES

of a right triangle using the

Pythagorean Theorem

The following are the basic steps for solving a

Pythagorean Theorem Problem.

Step 1: Write the formula

Step 2: Substitute known values for the

variables.

Step 3: Solve for the missing variable.

Lets break this down a little further…

12.
No need

Finding the missing side of a For notes

On this slide

right triangle

• Any time you are asked to find the missing

side of a right triangle, the problem will

generally boil down to 1 of 2 scenarios.

• Scenario 1: You have both legs and you have

to find the hypotenuse

• Scenario 2: You have one leg and the

hypotenuse, and you have to find the other

leg.

Finding the missing side of a For notes

On this slide

right triangle

• Any time you are asked to find the missing

side of a right triangle, the problem will

generally boil down to 1 of 2 scenarios.

• Scenario 1: You have both legs and you have

to find the hypotenuse

• Scenario 2: You have one leg and the

hypotenuse, and you have to find the other

leg.

13.
Scenario 1: Need the hypotenuse TAKE

Find x x NOTES

8 ft

• Step 1: Write the formula. 15 ft

a2 + b 2 = c 2

• Step 2: Substitute or “Plug-in” the lengths of the legs into the Pythagorean

Theorem for the “a” and “b” variables.

82 + 152 = c2

• Step 3: Simplify the side without the “c” by squaring the two numbers and adding them

together. We

2 are not done yet…

64 + 225 = c We have found c , but not 2

289 = c2 just plain c.

• Step 4: Solve for c by using the square root.

We were told to solve for x,

289 = c2 not c, so we should replace th

17 = c c with an x. x=

Find x x NOTES

8 ft

• Step 1: Write the formula. 15 ft

a2 + b 2 = c 2

• Step 2: Substitute or “Plug-in” the lengths of the legs into the Pythagorean

Theorem for the “a” and “b” variables.

82 + 152 = c2

• Step 3: Simplify the side without the “c” by squaring the two numbers and adding them

together. We

2 are not done yet…

64 + 225 = c We have found c , but not 2

289 = c2 just plain c.

• Step 4: Solve for c by using the square root.

We were told to solve for x,

289 = c2 not c, so we should replace th

17 = c c with an x. x=

14.
No need

Scenario 1 For notes

On this slide

What does all of this boil down to?

• Square both legs.

• Add them together.

• Take the square root of the result.

• You have your hypotenuse.

Scenario 1 For notes

On this slide

What does all of this boil down to?

• Square both legs.

• Add them together.

• Take the square root of the result.

• You have your hypotenuse.

15.
TAKE

You try this one in your notes. NOTES

x

Find x 5 ft

12 ft

52 + 122 = x2

25 + 144 = x2

169 = x2

• Answer: x = 13

You try this one in your notes. NOTES

x

Find x 5 ft

12 ft

52 + 122 = x2

25 + 144 = x2

169 = x2

• Answer: x = 13

16.
Scenario 2: Have TAKE

NOTES

Hypotenuse, need one leg

Find x.

14 in x

Round to the nearest

6 in

• Can we do this the same way we did

the other example?

• Not exactly the same way, but similar.

• Let’s start this one the same way we did

the other ones and see what happens…

NOTES

Hypotenuse, need one leg

Find x.

14 in x

Round to the nearest

6 in

• Can we do this the same way we did

the other example?

• Not exactly the same way, but similar.

• Let’s start this one the same way we did

the other ones and see what happens…

17.
Scenario 2: Have Hypotenuse, need one leg

Find x.

Round to the nearest 14 in x

• Step 1: Write the formula. a + b = c

2 2 2

• Step 2: Substitute or “Plug-in” the lengths of the legs …

But we don’t have both legs…

6 in

• Here is where we have to do something a little different. We have to plug

in the hypotenuse and one of the legs.

Which number goes where?

You need to identify the hypotenuse. It’s the one opposite of

the right angle.

The hypotenuse is always going to be c. So, the c = 14.

We need one more variable replaced in order to solve for the

missing variable. So, we need to replace either a or b with

the one leg length we have, which is 6.

Does it matter whether we use a = 6 or b = 6? No.

Let’s set b = 6 and make a the missing length

Find x.

Round to the nearest 14 in x

• Step 1: Write the formula. a + b = c

2 2 2

• Step 2: Substitute or “Plug-in” the lengths of the legs …

But we don’t have both legs…

6 in

• Here is where we have to do something a little different. We have to plug

in the hypotenuse and one of the legs.

Which number goes where?

You need to identify the hypotenuse. It’s the one opposite of

the right angle.

The hypotenuse is always going to be c. So, the c = 14.

We need one more variable replaced in order to solve for the

missing variable. So, we need to replace either a or b with

the one leg length we have, which is 6.

Does it matter whether we use a = 6 or b = 6? No.

Let’s set b = 6 and make a the missing length

18.
Scenario 2: Have Hypotenuse, need one leg

Find x.

Round to the nearest 14 in x

• Step 1: Write the formula. a 2

+ b2 = c 2

• Step 2: Identify the hypotenuse

6 in

• Step 3: Substitute or “Plug-in” the hypotenuse (14) for c and the other

known measurement (6) for b.

a2 + 62 = 142

• Step 4: Simplify by squaring both the numbers.

a2 + 36 = 196

At this point, in the previous example, we added the two squares

together. This time, the squares are on opposite sides of the equals

sign. So, to combine them, we have to do the opposite operation.

• Step 5: Subtract the smallera2 + 36 = 196

from the larger. – 36 – 36

a2 = 160

Find x.

Round to the nearest 14 in x

• Step 1: Write the formula. a 2

+ b2 = c 2

• Step 2: Identify the hypotenuse

6 in

• Step 3: Substitute or “Plug-in” the hypotenuse (14) for c and the other

known measurement (6) for b.

a2 + 62 = 142

• Step 4: Simplify by squaring both the numbers.

a2 + 36 = 196

At this point, in the previous example, we added the two squares

together. This time, the squares are on opposite sides of the equals

sign. So, to combine them, we have to do the opposite operation.

• Step 5: Subtract the smallera2 + 36 = 196

from the larger. – 36 – 36

a2 = 160

19.
Scenario 2: Have Hypotenuse, need one leg

Find x.

Round to the nearest 14 in x

• Step 1: Write the formula. a 2

+ b2 = c 2

• Step 2: Identify the hypotenuse

6 in

• Step 3: Substitute or “Plug-in” the hypotenuse (14) for c and the other

known measurement (6) for b.

a2 + 62 = 142

• Step 4: Simplify by squaring both the numbers.

a2 + 36 = 196

• Step 5: Subtract the smaller from the larger.

a2 + 36 = 196

• Step 6: Solve for a by using the

– 36 – 36

square root. a2 = 160

a2 = 160

a=

a = 12.64911

Find x.

Round to the nearest 14 in x

• Step 1: Write the formula. a 2

+ b2 = c 2

• Step 2: Identify the hypotenuse

6 in

• Step 3: Substitute or “Plug-in” the hypotenuse (14) for c and the other

known measurement (6) for b.

a2 + 62 = 142

• Step 4: Simplify by squaring both the numbers.

a2 + 36 = 196

• Step 5: Subtract the smaller from the larger.

a2 + 36 = 196

• Step 6: Solve for a by using the

– 36 – 36

square root. a2 = 160

a2 = 160

a=

a = 12.64911

20.
No need

Scenario 2 For notes

On this slide

What does all of this boil down to?

• Square the hypotenuse and leg.

• Subtract the leg squared from the

hypotenuse squared.

• Take the square root of the result.

• You have your missing leg.

Scenario 2 For notes

On this slide

What does all of this boil down to?

• Square the hypotenuse and leg.

• Subtract the leg squared from the

hypotenuse squared.

• Take the square root of the result.

• You have your missing leg.

21.
What is the difference between For

No need

notes

the 2 scenarios? On this slide

• Both have you squaring the given sides.

• Both have you using the square root at the

end.

• The only difference is in the middle.

• Scenario 1 has you adding the numbers

• Scenario 2 has you subtracting the smaller

from the larger.

No need

notes

the 2 scenarios? On this slide

• Both have you squaring the given sides.

• Both have you using the square root at the

end.

• The only difference is in the middle.

• Scenario 1 has you adding the numbers

• Scenario 2 has you subtracting the smaller

from the larger.

22.
What does this mean?

• When you have two sides of a right triangle,

you can find the third using the Pythagorean

Theorem.

• You can do this by squaring both of the

measurements you have.

• Add or subtract the two numbers depending

on whether or not you have the hypotenuse.

(Subtract if you have it, add if you don’t)

• Find the square root of the result and you

have your missing side!

• When you have two sides of a right triangle,

you can find the third using the Pythagorean

Theorem.

• You can do this by squaring both of the

measurements you have.

• Add or subtract the two numbers depending

on whether or not you have the hypotenuse.

(Subtract if you have it, add if you don’t)

• Find the square root of the result and you

have your missing side!

23.
Try this one in your notes…

x

15

20

Solve for x.

Round your answer to the nearest hundredth if necessary.

Answer: 25

x

15

20

Solve for x.

Round your answer to the nearest hundredth if necessary.

Answer: 25

24.
Try this one in your notes…

12 7

x

Solve for x.

Round your answer to the nearest hundredth if necessary.

Answer: 13.89

12 7

x

Solve for x.

Round your answer to the nearest hundredth if necessary.

Answer: 13.89

25.
Try this one in your notes…

5

x

3

Solve for x.

Round your answer to the nearest hundredth if necessary.

Answer: 4

5

x

3

Solve for x.

Round your answer to the nearest hundredth if necessary.

Answer: 4

26.
Try this one in your notes…

30

7

x

Solve for x.

Round your answer to the nearest hundredth if necessary.

Answer: 29.17

30

7

x

Solve for x.

Round your answer to the nearest hundredth if necessary.

Answer: 29.17