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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. Warm-Up Solve for x 1. x2+7=43 2. 64+x2=164 Ans: x = ±6 Ans: x = ±10 Evaluate for a = 12, b = 5, c = 13 Ans: 169 2 2 3. a + b Ans: 144 2 2 4. c – b Essential ?? • How can we use the Pythagorean Theorem to solve for a missing length of a right triangle? Pythagorean Theorem No need For notes On this slide • What is the Pythagorean Theorem in symbol form? 2 a + 2 b 2 c = • 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 Steps to Solve for a missing side TAKE of a right triangle using the NOTES Pythagorean Theorem Step 1: Write the formula Step 2: Substitute known values for the variables. Step 3: Solve the equation for the missing variable. Example 1 Find x 8 ft x TAKE NOTES 15 ft • Step 1: Write the formula 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. Does it matter whether we use a = 8 or 15? No. Let’s use a = 8 and b = 15. Example 1 Find x TAKE NOTES x 8 ft 15 ft • Step 1: Write the formula 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… We have found c2, but not just plain c. We were told to solve for x, not c. So we should replace the c with an x. 64 + 225 = c2 289 = c2 289 = c2 17 = c x = 17 You try this one in your notes. Find x 5 ft x 12 ft 52 + 122 = x2 25 + 144 = x2 169 = x2 • Answer: x = 13 TAKE NOTES TAKE NOTES x Example #2 Find x. Round to the nearest hundredth. 14 in 6 in • Step 1: Write out the formula • Step 2: Substitute known values a2 + b2 = c2 a2 + 62 = 142 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? Let’s use b = 6. No TAKE NOTES Example #2 Find x. Round to the nearest hundredth. 14 in x 6 in a2 + b2 = c2 a2 + 62 = 142 • Step 1: Write out the formula • Step 2: Substitute known values • Step 3: Solve for the missing variable, in this case a. Can we just add the two numbers and do the square root? No, they are not on the same side of the equals sign. x = 12.65 a2 + 36 = 196 – 36 – 36 a2 = 160 a2 = 160 a = 12.64911 What is the difference between No need For notes On this slide 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. 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! Try this one in your notes… x 15 20 Solve for x. Round your answer to the nearest hundredth if necessary. Answer: 25 Try this one in your notes… 7 12 x Solve for x. Round your answer to the nearest hundredth if necessary. Answer: 13.89 Try this one in your notes… 5 x 3 Solve for x. Round your answer to the nearest hundredth if necessary. Answer: 4 Try this one in your notes… 30 7 x Solve for x. Round your answer to the nearest hundredth if necessary. Answer: 30.81 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? a) 6 + b = 10 b) 36 + b = 100 c) b2 – 36 = 100 d) 36 + b2 = 100 Answer: D