Pythagorean Theorem

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Transcript 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 Funke 8 th Grade Math Teacher Central Middle School West Melbourne, FL

Bell Ringer

• • Solve for x x 2 +7=43 64+x 2 =164

Ans: x = Ans: x = ±6 ±10

Evaluate for a = 12, b = 5, c = 13 3. a 2 + b 2 4. c 2 – b 2

Ans: 169 Ans: 144

Today’s Objective

No need For notes On this slide • 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.

Pythagorean Theorem

No need For notes On this slide • What is the Pythagorean Theorem in symbol form?

a 2 + b 2 = c 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

Steps to Solve for a missing Side of a triangle using the Pythagorean Theorem

TAKE NOTES

Step 1: Write the formula Step 2: Substitute known values for the variables.

Step 3: Solve the equation for the missing variable.

Find x

Example 1

8 ft x TAKE NOTES

Step 1:

Write the formula

15 ft a 2 + b 2 = c 2

Step 2:

Substitute known values

8 2 + 15 2 = c 2

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 the c in the formula. Since we do not know the value of c, it stays as c in the formula.

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

Find x

Example 1

8 ft x TAKE NOTES

Step 1:

Write the formula

15 ft a 2 + b 2 = c 2

• •

Step 2:

Substitute known values

8 2 + 15 2 = c 2 Step 3:

Solve for the missing variable, in this case c.

We are not done yet… We have found c 2 , but not just plain c.

We were told to solve for x, not c. So we should replace the c with an x. x = 17 64 + 225 = c 2 289 = c 2 289 = c 2 17 = c

You try this one in your notes.

TAKE NOTES x Find x 5 ft 12 ft 5 2 + 12 2 = x 2 25 + 144 = x 2 169 = x 2

• Answer:

x = 13

Example #2

TAKE NOTES

• •

14 in x Find x.

Round to the nearest hundredth.

6 in Step 1:

Write out the formula

a 2 + b 2 = c 2 a 2 Step 2:

Substitute known values Which number goes where? This time we are given the hypotenuse. So, c = 14

+ 6 2 = 14 2

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

Example #2

TAKE NOTES 14 in x Find x.

Round to the nearest hundredth.

6 in

• • •

Step 1: Step 2:

Write out the formula Substitute known values

a a 2 2 + b + 6 2 2 = c 2 = 14 2 Step 3:

Solve for the missing variable, in this case a.

Can we just add the two numbers and do the square root? No, they are not on

a 2 + 36 = 196 – 36 – 36

the same side of the equals sign.

a 2 = 160 x = 12.65

a 2 = 160 a = 12.64911

What is the difference between the 2 examples?

No need For notes 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.

– Example 1 has you adding the numbers – Example 2 has you subtracting the smaller from the larger.

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…

x 15 20 Solve for x. Round your answer to the nearest hundredth if necessary.

Answer: 25

Try this one in your notes…

7 12 x Solve for x. Round your answer to the nearest hundredth if necessary.

Answer: 13.89

Try this one in your notes…

5 x 3 Solve for x. Round your answer to the nearest hundredth if necessary.

Answer: 4

Try this one in your notes…

30 7 Solve for x. x Round your answer to the nearest hundredth if necessary.

Answer: 30.81

Try this one in your notes…

b a c If the hypotenuse of this triangle is 10 and

a

is 6 which equation would you use to find b?

a) 6 + b = 10 b) 36 + b = 100 c) b 2 – 36 = 100 d) 36 + b 2 = 100

Answer: D