1. Pythagorean Theorem
CC A.3 - Apply the Pythagorean Theorem to determine
unknown side lengths in right triangles in real-
world and mathematical problems in two and
three dimensions.
MATH 8

2. Bell Ringer Solve for x x2+7=43 64+x2=164
Evaluate for a = 12, b = 5, c = 13
a2 + b2
c2 – b2

3. Objective We are going to learn about the Pythagorean Theorem.
We are going to learn how to use
the Pythagorean Theorem to solve for a
missing length of a right triangle.

5. a2 + b2 = c2 c no Pythagorean Theorem
What is the Pythagorean Theorem in symbol form?
a2 + b2 = c2
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

6. 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.

7. Example 1 x Find x 8 ft 15 ft a2 + b2 = c2 82 + 152 = c2
Step 1: Write the formula
a2 + b2 = c2
Step 2: Substitute known values
= c2
Which number goes where?
You need to identify the hypotenuse. (It’s the one opposite of the right angle.)
The hypotenuse is always the c in the formula. We do not know the value of c, so 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.

8. Example 1 x Find x 8 ft 15 ft a2 + b2 = c2 82 + 152 = c2 64 + 225 = c2
Step 1: Write the formula
a2 + b2 = c2
Step 2: Substitute known values
= c2
Step 3: Solve for the missing variable, in this case c.
We are not done yet…
= c2
We have found c2, but not just plain c.
289 = c2
289 = c2
We were told to solve for x, not c.
So we should replace the c with an x.
17 = c
x = 17

9. YOU TRY x Find x 5 ft 12 ft 52 + 122 = x2 25 + 144 = x2 169 = x2
Answer:
x = 13

10. Example #2 14 in x Find x. 6 in a2 + b2 = c2 a2 + 62 = 142
Round to the nearest
hundredth.
6 in
a2 + b2 = c2
Step 1: Write out the formula
a = 142
Step 2: Substitute known values
Which number goes where?
This time we are given the hypotenuse.
So, c = 14
Does it matter whether we use a = 6 or b = 6? No
Let’s use b = 6.

11. Example #2 14 in x Find x. 6 in a2 + b2 = c2 a2 + 62 = 142
Round to the nearest
hundredth.
6 in
a2 + b2 = c2
Step 1: Write out the formula
a = 142
Step 2: Substitute known values
Step 3: Solve for the missing variable.
a = 196
Can we just add the two numbers and do the square root?
– – 36
No, they are not on the same side of the equals sign.
a2 = 160
a2 =
x = 12.65
a =

12. What is the difference between the 2 examples?
Both have us squaring the given sides.
Both have us using the square root at the end.
The only difference is in the middle.
Example 1 has us adding the numbers
Example 2 has us subtracting the smaller from the larger.

13. 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!

14. Try this one. Answer: x 15 20 Solve for x.
Round your answer to the nearest hundredth if necessary.
Answer:

15. Try this one. Answer: 7 12 x Solve for x.
Round your answer to the nearest hundredth if necessary.
Answer:

16. Try this one. Answer: 5 x 3 Solve for x.
Round your answer to the nearest hundredth if necessary.
Answer:

17. Try this one in your notes…30 7 x
Solve for x.
Round your answer to the nearest hundredth if necessary.
Answer:

18. Try this one. Answer: b a c
If the hypotenuse of this triangle is 10 and a is 6 which equation would you use to find b?
6 + b = 10
36 + b = 100
b2 – 36 = 100
36 + b2 = 100
Answer: