The Pythagorean Theorem

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Presentation transcript:

The Pythagorean Theorem Lovin’ Those Right Triangles!

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.

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

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

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.

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

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

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

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

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.

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

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.