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5.8 SQUARE ROOTS & THE PYTHAGOREAN THEOREM Review: Given a square that has a side whose length is 12 cm, find the area. 12 cm If we know that the area of a square is 25 sq. yds., what is the length of each side? 25 sq yds In this second case, we look for a number that when you SQUARE it, you get 25. We say that number is the SQUARE ROOT of 25. The positive SQUARE ROOT of a positive number is a POSITIVE number that when multiplied TIMES ITSELF (squared), you will get that number. Ex: The square root of 25 is 5 since 5 x 5 = 25 Using symbols, we write 25 = 5 You should know the square roots of all “perfect squares” through 144 (lets generate that together, also on page A-2). For right now, when we do NOT have a perfect square, we will use a CALCULATOR to approximate the square root by pressing the number, then the square root key on your calculator, finally write your answer rounded to the nearest thousandth. Ex: approximate 27 When we press the correct keys on our calculator we get: _______________ … (the periods show it continues, but we don’t know how, the calculator has to stop somewhere), which we would round to 5.__________ . (Nearest thousandth.) (See section 15.2 to find and write square roots without using a calculator. THAT IS A VERY IMPORTANT SECTION FOR ALGEBRA!) Ex: Find the approximate square root of 125. Ex: Simplify 9 64 (If time.) Now we will use this to apply it to finding missing sides of a triangle using the PYTHAGOREAN THEOREM. (Would you like to see a proof?) The Pythagorean theorem states the relationship among the LENGTHS of the SIDES of a RIGHT triangle as: a2 + b2 = c2 a c b Where a and b are the lengths of the sides, and c is the length of the “hypotenuse” (side opposite the 90 degree angle). The hypotenuse is ALWAYS LONGEST. If we want to find the length of the hypotenuse given the other two sides, we use: hypotenuse = (leg a)2 + (leg b)2 If we want to find the length of one of the legs given one side and the hypotenuse we use: leg = (hypotenuse)2 - (other leg)2 Ex: Find the length of the missing side. (Remember you are finding just the length of a side, so the units will be to the _________power). 8 cm ? 6 cm 10 m 5m ? Application: Lax Larry took the scenic route to his first service call. From home he headed due south 30 miles, then took a sharp left and headed east for 90 miles. On the way home he took the street that went directly from the service call to home. How far was he actually from work if he takes the shortest route? (Draw a picture). (How many extra miles did he bill his service call for?) Add p 441 # 29 and 31 to your homework.