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Pythagorean Theorem Sarah Wales Introduction/present objective: Graphic: a right triangle with an arrow pointing at the hypotenuse Voice: In this lesson you will learn how to use the Pythagorean Theorem to find the length of a right triangle’s hypotenuse. First, let’s review some vocabulary. Vocabulary Review: Voice: This is the hypotenuse, and these other two sides are called legs. The hypotenuse is always the side opposite to the right angle. Introduce the formula a2 + b2 = c2 c a b Voice: The formula for the Pythagorean Theorem is a squared plus b squared equals c squared, where the legs are a and b and the hypotenuse is c. Animated Proof: Voice: Leg a squared, plus leg b squared equals the hypotenuse squared 2 a + 2 b = 2 c 2 a + 2 b = 2 c First example: Text animation: Animate dropping the 6 and 8 into the formula. Animate formula solution, squaring 6 and 8 then adding them to get 100. Animate finding the square root of 100. Animate moving the answer, 10, back to the hypotenuse on the graphic. c 6 8 Voice: Here is another right triangle. Leg a is 6 units long; leg b is 8 units long. How long is the hypotenuse? Six squared is 36 and 8 squared is 64. Thirty-six plus 64 is 100, so c squared equals 100. The square root of 100 is 10, therefore the hypotenuse is 10 units long. Second Example: c 12 5 Voice: Let’s solve one together. Maria wants to install a window box outside of her apartment, but the instructions require her to access the exterior of her window to securely bolt the window box to the wall. She lives on the second floor with her window ledge at 12 feet above the ground, so she will need a ladder. The ladder must be the right length or it won’t rest at a safe angle on her 5 foot wide sidewalk. How long does the ladder need to be? First, let’s put the values we know into the formula. Pause the video and see if you can do that on your own. Did you do it? I know that a equals 5 and b equals 12. If you wrote a is 12 and b is 5, you’re doing great! The order of a and b does not matter. Now let’s square each of the legs. Pause the video while you try it. Did you write 25 plus 144 equals c squared? 5 squared is 25 and 12 squared is 144. Now let’s add them together. Pause the video while you find the sum. Did you get 169? Great! If not, check your addition carefully. Now we know that c squared equals 169. What do we do next? Pause the video and try to find the answer on your own if you can. We need to find the square root of 169 to find the value of c, the length of the ladder Maria needs. The square root of 169 is 13. Maria needs a 13 foot ladder. How did you do? Review the video from the beginning if you do not feel confident yet. Next, you’ll try a problem on your own. Independent Practice: 12 9 ? Voice: here is another problem for you to solve. Milo is building a set of triangular raised garden beds out of cedar planks. He has already nailed together two planks at a right angle. One plank is 9 feet long and the other is 12 feet long. Now he needs to saw the third plank, but he isn’t sure where to make the cut. Can you help him? Pause the video now and see if you can solve this one on your own. If you get stuck, try looking at the steps we used to solve Maria’s ladder problem. Did you solve it? The answer is 15 feet. If you got a different answer, follow along with me while I solve it. First I substitute the values that I have for a and b. a is 9 and b is 12. If you put them in a different order, that is fine. Next, I square a and b. Nine squared is 81 and 12 squared is 144. Next I add a and b together. 81 plus 144 equals 225. Now I know that c squared is 225. To find out how long the cedar plank needs to be, I need the square root of that number. The square root of 225 is 15. Milo needs to cut the plank to 15 feet. Conclusion • Voice: That is how we can use the Pythagorean Theorem!