1. The Pythagorean Theorem
Lovin’ Those Right Triangles!

2. The Pythagorean Theorem
In most of the ancient civilizations that concerned themselves with mathematics, individuals discovered a relationship with sides of a right triangle.
Pythagoras was credited with the theorem, even though it was probably discovered long before him.

3. So What’s the Relationship?
In a right triangle with legs a and b, and hypotenuse c, c a b

4. Prove it!
Here are some visual proofs of the Pythagorean theorem: Let’s check ‘em out!

5. Using it to find the length of a hypotenuse
If you know the two legs, you can plug in the values for a and b.
Square both of them
Add them together
Take the square root of the sum.

6. Ex 6 8 A rectangle has sides of 6 and 8. How long is the diagonal?
A rectangle is just two right triangles put together right?
So for this problem, a = 6, b = 8, and we’ll let c = x. 6 8

7. Using the Formula
So if a = 6, b = 8, and c = x, when we substitute into , we get
*Now, to solve for x, we’ll square the 6 and 8.
*We can add the 36 and 64.
*Finally, to undo squaring, we take the square root of each side

8. Ex2
A right triangle has legs of length 3 and 4. What is the length of the hypotenuse? Remember to draw your figure!

9. Solution First we draw the figure: Now we find our a, b, and c.x
First we draw the figure:
Now we find our a, b, and c.
4 and 3 are legs, so they can be either a or b.
We’ll say a = 4, b = 3, and c = x
Substitute into the theorem and get:
Try solving it on your own!
You should get x = 5 4 3

10. Finding a missing leg This is slightly harder
First, plug in c and either a or b.
Then square them.
You’ll then have to subtract to get the missing leg by itself
Finally, take the square root.

11. Ex 25 cm 27 cm a Find the length of the missing side:
For this problem, a = x, b = 25, & c = 27
Substitute them into and get
*square the 25 and 27
*To solve for x, subtract the 625
*Now take the square root Since it isn’t perfect, round to the tenths place a

12. Ex 2
If a right triangle has a hypotenuse of 14 in and one of the legs is 9 in, find the third leg.
Try this one on your own.
Start by drawing the figure.
Set a = x, b = 9, and c = 14.
Solve it like the last problem.
You should get x = 10.7 in.