1. Using the Pythagorean Theorem Sarah Katko ICL 7062

2. next slide The Pythagorean Theorem is the formula used to find the missing side of a right triangle. The purpose of this tutorial is to thoroughly explain/illustrate how to solve problems through the application of this theorem. There will be opportunities for you to practice this skill throughout this module. You may use the navigation bar at on the left of this home screen to jump to specific sections of this tutorial. To access the bar throughout the tutorial, simply press the home button in the bottom of any screen to return to this page. When you are finished with this tutorial, you will demonstrate the ability to solve for the missing side of a right triangle with at least 80% mastery. home previous slide

3. The formula for the Pythagorean Theorem is: We can use this formula to find the length of a missing side of a right triangle. The formula for the Pythagorean Theorem is: We can use this formula to find the length of a missing side of a right triangle. A right triangle is made up of three sides: two legs and a hypotenuse. “A” represents one leg of the right triangle, and ”B” represents the other leg. The third side is the hypotenuse, and it is the longest side. The hypotenuse is represented by the letter “C” in the formula. A right triangle is made up of three sides: two legs and a hypotenuse. “A” represents one leg of the right triangle, and ”B” represents the other leg. The third side is the hypotenuse, and it is the longest side. The hypotenuse is represented by the letter “C” in the formula. home next slide

4. home previous slide next slide It is imperative to correctly identify the sides of the triangle, so that the numbers are in the correct place in the formula. While the legs are interchangeable (it does not matter if you switch “A” and “B” around), the hypotenuse must always be “C”. ***The hypotenuse is ALWAYS the longest side in the right triangle.**** A helpful hint for determining the hypotenuse is to locate the right angle, which is marked and indicated by a small box. Use the right angle symbol there to create an arrow. The arrow will always point to the hypotenuse! It is imperative to correctly identify the sides of the triangle, so that the numbers are in the correct place in the formula. While the legs are interchangeable (it does not matter if you switch “A” and “B” around), the hypotenuse must always be “C”. ***The hypotenuse is ALWAYS the longest side in the right triangle.**** A helpful hint for determining the hypotenuse is to locate the right angle, which is marked and indicated by a small box. Use the right angle symbol there to create an arrow. The arrow will always point to the hypotenuse!

5. previous slide next slide Now that we have the basic information, lets see how this works. In the right triangle below, all sides are measured, so we will use the Pythagorean Theorem to check for accuracy. We will use our formula to plug in numbers correctly. First, I can identify my legs (A and B) as 3 and 4. The hypotenuse (longest side) is labeled 5, and I can draw an arrow in my right angle symbol to make sure it is indeed the hypotenuse. Now, I will plug my legs in for A and B and my hypotenuse in for C. a 2 +b 2 = c 2 3 2 + 4 2 = 5 2 Next, I will square the numbers a 2 ( 3 x 3) = 9 b 2 ( 4 x 4) = 16 c 2 (5 x 5) = 25 Last, I will plug the numbers into the formula and make sure both sides are equal. a 2 +b 2 = c 2 9 + 16 = 25 ✔ Since both sides are equal, we know this is correct! home

6. previous slide next slide a 2 +b 2 = c 2 therefore 3 2 +4 2 = c 2 Now, we will square the numbers and plug them back into the formula. 3 2 = 9 and 4 2 = 16 9 + 16 = c 2 9 + 16 = 25 therefore 25 = c 2 If c 2 is 25, and we want to solve for C, we must undo the square by doing the opposite. We must find the square root of 25. √25 = 5, therefore 5 is C Just to be sure, let’s check: a 2 +b 2 = c 2 3 2 +4 2 = 5 2 9 + 16 = 25 ✔ Both sides are equal, so you know you are correct! Let’s apply what we have just learned to solve for C, the hypotenuse. In the picture below, we can see that the point of the arrow is indeed pointing to the longest side, which is missing its measurement. Therefore, we know that we are solving for C. Our legs are labeled with numbers to be placed in the formula for the Pythagorean Theorem. home

7. previous slide next slide A. 10 feet B. 14 feet C. 25 Feet D. 100 feet Let’s try a contextual problem with a little guidance. Your sides have been identified for you, so plug them into the formula first. Remember: a 2 +b 2 = c 2 home Side “A” is 8 Side “B” is 6 Square sides A and B, then add them together. Find the square root of the total, and that is the length of side C! Once you get your answer, proceed to the next slide for an explanation.

8. previous slide next slide home A. 10 feet B. 14 feet C. 25 Feet D. 100 feet If you got A, 10 feet for your answer, you are correct! Explanation: a 2 +b 2 = c 2 The branches are leaning against the wall, which is side C. The ground where the sleeping bag is 6 feet long, and the wall it is against is 8 feet tall. These numbers represent the legs, A and B. Steps: Plug in the sides : 6 2 + 8 2 =c 2 Square and add together: 36 + 64 = c 2 Add A and B together: 36 + 64 = 100 Since 100 represents c 2 we must do the opposite or inverse of squaring, which is finding the square root. √100 = 10 Check: 6 2 + 8 2 = 10 2 36 + 64 = 100 ✔ Both sides are equal, so it balances the equation! Hint: looking at the answer choices, we can automatically rules out choice D. This is way too large of a number if one side is 6 and the other is 8. Test writers expect students to choose this answer because it is the in the process, but not the final answer! Be careful!

9. previous slide next slide A) 16.7 units B) 4.5 units C) 8.9 units D) 14.4 units A) 16.7 units B) 4.5 units C) 8.9 units D) 14.4 units It’s time to do one on your own! If you need help with the steps, press the button to return to the previous slide. HINT: since you are solving for the longest side, there are two answers that can automatically be ruled out. Once you have solved for side c, click on the answer you have chosen for feedback. home

10. previous slide next slide A) 10.23 feet B) 16.09 feet C) 23.59 feet D) 26.63 feet A) 10.23 feet B) 16.09 feet C) 23.59 feet D) 26.63 feet Time to try one more before moving on to solving for a missing leg! If you are still feeling confused, or would just like additional information, please click on the link to watch the video for further explanation: khan academy pythagorean theorem. The last four minutes of the video will lead you into the next part of our lesson.khan academy pythagorean theorem home

11. previous slide next slide home Now that we have used the Pythagorean Theorem to solve for a missing hypotenuse, will move on to solving for a missing leg. To do this we will begin the same way, by plugging what we have into the formula: a 2 +b 2 = c 2 26 2 + b 2 = 52 2 Now, square the numbers and plug the answers back into the formula: 26 2 = 676 and 52 2 = 2704 therefore 676 + b 2 = 2704 Now we will apply what we know about solving equations (performing inverse operations ) to get “b” by itself. 676 + b 2 = 2704 -676 b 2 = 2028 The “b” is still squared, so we must find the square root of each side to get it alone. √ b = √2028 therefore b= 45 mm Now that we have used the Pythagorean Theorem to solve for a missing hypotenuse, will move on to solving for a missing leg. To do this we will begin the same way, by plugging what we have into the formula: a 2 +b 2 = c 2 26 2 + b 2 = 52 2 Now, square the numbers and plug the answers back into the formula: 26 2 = 676 and 52 2 = 2704 therefore 676 + b 2 = 2704 Now we will apply what we know about solving equations (performing inverse operations ) to get “b” by itself. 676 + b 2 = 2704 -676 b 2 = 2028 The “b” is still squared, so we must find the square root of each side to get it alone. √ b = √2028 therefore b= 45 mm Side C Side A Side B

12. previous slide next slide. A) 6 B) 8 C) 9 D) 10 A) 6 B) 8 C) 9 D) 10 home Solve the problem and click on the answer you have selected for feed back. Use the following steps to try it on your own: Plug numbers into formula, paying close attention to identifying the hypotenuse (C). Square the numbers and plug back into formula. Solve the equation by performing inverse operations. Do not forget to find the square root at the end!

13. previous slide next slide A ) 9 units B) 20 units C) 40 units D) 80 units A ) 9 units B) 20 units C) 40 units D) 80 units home Solve for the missing leg and click your answer for feedback. Feel free to visit the previous slides for help. Hint!

14. previous slide next slide A) 3.2 feet B) 10.5 feet C) 33.2 feet D) 78.1 feet A) 3.2 feet B) 10.5 feet C) 33.2 feet D) 78.1 feet home Solve for the missing leg and click your answer for feedback.

15. home previous slide next slide Congratulations, you have successfully completed the module! For additional practice with the Pythagorean Theorem, click here: practice problems If you are ready to take an assessment on the Pythagorean Theorem click here : Discovery Log InDiscovery Log In Enter your last name, leave a space, and enter your first name. Enter this code: PZPUQ203087 You will get to choose a game to play for correct answers and feedback will be given. Good luck!

16. Awesome work! You are correct! Click below to return to the last slide viewed and then move on to the next slide.

17. Press the button to return to the slide and try again. Press the button to return to the slide and try again. Make sure you have plugged everything in correctly by using the diagram. After squaring the numbers and adding them together, find the square root of the total.

18. Remember the steps! Plug numbers into formula, paying close attention to identifying the hypotenuse (C). DO YOU HAVE NUMBERS ON BOTH SIDES OF THE EQUATION? Square the numbers and plug back into formula. Solve the equation by performing inverse operations. THIS MEANS SUBTRACT FROM BOTH SIDES! Do not forget to find the square root at the end! Remember the steps! Plug numbers into formula, paying close attention to identifying the hypotenuse (C). DO YOU HAVE NUMBERS ON BOTH SIDES OF THE EQUATION? Square the numbers and plug back into formula. Solve the equation by performing inverse operations. THIS MEANS SUBTRACT FROM BOTH SIDES! Do not forget to find the square root at the end! Press the button to return to the slide and try again. Press the button to return to the slide and try again.