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

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

3. 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

4. Steps to Solve for a Missing
Side of a 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.

5. Example 1
Step 1: Write the formula
c
8 ft
15 ft
Find
c
a2 + b2 = c2
82 + 152 = c2Step 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.

6. Example 1
Step 1: Write the formula
x
8 ft
15 ft
Find
x
a2 + b2 = c2
82 + 152 = c2Step 2: Substitute known value
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 = c

7. Try this one in your notes…..
x
5 ft
12 ft
Find x

8. Di
52 + 122 = x2
25 + 144 = x2
169 = x2
Answer x = 13
Did your work
look like this?

9. Example #2
x14 in
6 in
Find x.
Round to the nearest
hundredth.
Step 1: Write out the formula a2 + b2 = c2
a2 + 62 = 142Step 2: Substitute known values
Which 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

10. Example #2
x14 in
6 in
Find x.
Round to the nearest
hundredth.
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 = 196
Can we just add the two numbers and take the
square root? No, they are not on the same
side of the equals sign.
– 36 – 36
a2 = 160
a2 = 160
a = 12.64911
x = 12.65

11. 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.

12. 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!

13. Try this one in your notes…
15
20
x
Solve for x.
Round your answer to the nearest hundredth if necessary.
Answer: 25

14. Try this one in your notes…
7
12
x
Solve for x.
Round your answer to the nearest hundredth if necessary.
Answer: 13.89

15. Try this one in your notes…
3
x5
Answer: 4
Solve for x.
Round your answer to the nearest hundredth if necessary.

16. Try this one in your notes…
7
x
30
Answer: 30.81
Solve for x.
Round your answer to the nearest hundredth if necessary.

17. Try this one in your notes…
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 = 10
b) 36 + b = 100
c) b2 – 36 = 100
d) 36 + b2 = 100

18. Please email me or call
me if you have any
difficulties working the
practice problems.