Pythagorean Theorem

1. Pythagorean Theorem MCC8.G.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

2. Warm-Up Ans: x = ±6 Solve for x • x2+7=43 • 64+x2=164 Evaluate for a = 12, b = 5, c = 13 • a2 + b2 • c2 – b2 Ans: x = ±10 Ans: 169 Ans: 144

3. Essential ?? • How can we use the Pythagorean Theorem to solve for a missing length of a right triangle?

5. No need For notes On this slide 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

6. Steps to Solve for a missing side of a right triangle using the Pythagorean Theorem TAKE NOTES Step 1: Write the formula Step 2: Substitute known values for the variables. Step 3: Solve the equation for the missing variable.

7. TAKE NOTES Example 1 x Find x 8 ft • Step 1: Write the formula 15 ft a2 + b2 = c2 • Step 2: Substitute known values 82 + 152 = c2 Which number goes where? You need to identify the hypotenuse. It’s the one opposite of the right angle. 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. No. Does it matter whether we use a = 8 or 15? Let’s use a = 8 and b = 15.

8. TAKE NOTES Example 1 x Find x 8 ft • Step 1: Write the formula 15 ft a2 + b2 = c2 • Step 2: Substitute known values 82 + 152 = c2 • Step 3: Solve for the missing variable, in this case c. We are not done yet… 64 + 225 = c2 We have found c2, but not just plain c. 289 = c2 289 = c2 We were told to solve for x, not c. So we should replace the c with an x. 17 = c x = 17

9. TAKE NOTES You try this one in your notes. x Find x 5 ft • Answer: 12 ft 52 + 122 = x2 25 + 144 = x2 169 = x2 x = 13

10. TAKE NOTES Example #2 14 in x Find x. Round to the nearest hundredth. • Step 1: Write out the formula 6 in a2 + b2 = c2 a2 + 62 = 142 • Step 2: Substitute known values Which number goes where? This time we are given the hypotenuse. So, c = 14 Does it matter whether we use a = 6 or b = 6? No Let’s use b = 6.

11. TAKE NOTES Example #2 14 in x Find x. Round to the nearest hundredth. • Step 1: Write out the formula 6 in 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 do the square root? – 36 – 36 No, they are not on the same side of the equals sign. a2 = 160 a2 = 160 x = 12.65 a = 12.64911

12. No need For notes On this slide 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.

13. 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!

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

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

16. Try this one in your notes… 5 x 3 Solve for x. Round your answer to the nearest hundredth if necessary. Answer: 4

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

18. Try this one in your notes… b a c If the hypotenuse of this triangle is 10 and a is 6 which equation would you use to find b? • 6 + b = 10 • 36 + b = 100 • b2 – 36 = 100 • 36 + b2 = 100 Answer: D