Pythagorean Theorem

Pythagorean Theorem

• MACC.8.G.2.7 - Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Created by:Matthew Funke8th Grade Math TeacherCentral Middle SchoolWest Melbourne, FL

Bell Ringer

Solve for x

• x2+7=43

• 64+x2=164

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

3. a2 + b2

4. c2 – b2

Ans: x = ±6Ans: x = ±10

Ans: 169Ans: 144

Today’s Objective• We are going to learn more about the

Pythagorean Theorem.

• Today, we are going to learn how to use the Pythagorean Theorem to solve for a missing length of a right triangle.

No needFor notes

On this slide

Pythagorean Theorem• What is the Pythagorean Theorem in

symbol form?

a2 + b2 = c2

• Which of these variables represent the hypotenuse? c

No needFor notes

On this slide

• Once you have figured out which is c, does it matter which leg is a and which is b? no

Steps to Solve for a missing side of a right triangle using the

Pythagorean Theorem

Step 1: Write the formula

Step 2: Substitute known values for the variables.

Step 3: Solve the equation for the missing variable.

TAKENOTES

Example 1

• Step 1: Write the formula

TAKENOTES

x8 ft

15 ft

Find x

a2 + b2 = c2

82 + 152 = c2• Step 2: Substitute known values

Which number goes where? You need to identify the hypotenuse. It’s the one opposite of the right angle.

Does it matter whether we use a = 8 or 15? No. Let’s use a = 8 and b = 15.

The hypotenuse is always going to be the c in the formula. Since we do not know the value of c, it stays as c in the formula.

Example 1

• Step 1: Write the formula

TAKENOTES

x8 ft

15 ft

Find x

a2 + b2 = c2

82 + 152 = c2• Step 2: Substitute known values

64 + 225 = c2

289 = c2

We are not done yet… We have found c2, but not just plain c.

• Step 3: Solve for the missing variable, in this case c.

289 = c2

17 = cWe were told to solve for x, not c. So we should replace the c with an x.

x = 17

You try this one in your notes.

• Answer:

x5 ft

12 ft

Find x

52 + 122 = x2

25 + 144 = x2

169 = x2

x = 13

TAKENOTES

Example #2TAKE

NOTES

x14 in

6 in

Find x.Round to the nearesthundredth.

• Step 1: Write out the formula a2 + b2 = c2

a2 + 62 = 142• Step 2: Substitute known valuesWhich number goes where? This time we are given the hypotenuse.

Does it matter whether we use a = 6 or b = 6?

Let’s use b = 6.

So, c = 14

No

Example #2TAKE

NOTES

x14 in

6 in

Find x.Round to the nearesthundredth.

• Step 1: Write out the formula a2 + b2 = c2

a2 + 62 = 142• Step 2: Substitute known values• Step 3: Solve for the missing variable, in this case a.

a2 + 36 = 196Can we just add the two numbers and do the square root? No, they are not on the same side of the equals sign.

– 36 – 36 a2 = 160

a2 = 160a = 12.64911

x = 12.65

What is the difference between the 2 examples?

• Both have you squaring the given sides.

• Both have you using the square root at the end.

• The only difference is in the middle.– Example 1 has you adding the numbers– Example 2 has you subtracting the smaller from

the larger.

No needFor notes

On this slide

What does this mean?

• When you have two sides of a right triangle, you can find the third using the Pythagorean Theorem.

• Square 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!

Try this one in your notes…

15

20

x

Solve for x. Round your answer to the nearest hundredth if necessary.

Answer: 25


712

x


Answer: 13.89


3

x5

Answer: 4



7

x

30

Answer: 30.81



ab

c

Answer: D

If the hypotenuse of this triangle is 10 and a is 6 which equation would you use to find b?

a) 6 + b = 10b) 36 + b = 100c) b2 – 36 = 100d) 36 + b2 = 100

Pythagorean Theorem

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