The area of the larger square is: IF it s a right triangle, THEN + =


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1 8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht s omplited wy to sy tht if the legs of the tringle mesure nd nd the hypotenuse mesures, then + =. While you my hve herd this in the pst, we will now prove it. Proof of the Pythgoren Theorem There re mny wys to prove the Pythgoren Theorem, ut tke look t the following piture. We will refer to this for our proof. In this piture we hve lrge squre whose side lengths re equl to + nd n inner squre whose side lengths re. Notie tht if we find the re of the lrge squre nd sutrt the re of the tringles we get the re of the inner squre. So let s do tht lgerilly. The re of the lrger squre is: (+) =(+)(+)=(+)+(+)= +++ = +2+ The re of eh tringle is nd sine there re four of them, the totl re of the tringles is 2. The re of the inner squre is. This mens the lrge squre minus the tringles would look like this: +2+ 2= Notie tht the +2 nd the 2 nel eh other out (eome zero), so we do get the result we expet whih is tht + =. Do serh online to see if you n find nother proof for this vitl theorem. One more time, the IFTHEN sttement for the Pythgoren Theorem is: IF it s right tringle, THEN + = Sine we know tht in right tringle the sttement + = must e true, we n now solve for ny missing side length given the other two side lengths. The proess of solving for missing leg ( or ) is only slightly different from solving for missing hypotenuse (). 289
2 Solving for Missing Leg Let s first solve for missing leg. First note tht it mkes no differene whih leg we lel s nd whih leg we lel s. This is euse the ommuttive property sys tht we n dd in ny order. In other words, whether we hve + or + doesn t mtter, it will lwys equl. So if we re missing the length of leg, it might e esiest to lwys ssume it is tht is missing. 12 in. 13 in. Given the ft tht this is right tringle, we n solve for the missing leg length,. Just sustitute everything we know into the Pythgoren Formul. We know tht the hypotenuse length,, is 13 inhes nd tht the other leg length,, is 12 inhes. + = +(12) =(13) Now go hed nd multiple out those exponents to get the following sttement: +144=169 Notie this is twostep eqution where is eing squred nd then inresed y 144. Applying inverse opertions, we know we should sutrt 144 from oth sides nd then tke the squre root. Tht looks like this: 144= =25 = 25 =5 We hve just proved tht the missing side length must e 5 inhes. 290
3 Sometimes the missing side length will e leled with different vrile just to throw us off. Just rememer tht the legs re lwys nd in the Pythgoren Formul nd tht, or the hypotenuse, is lwys the longest side length. For exmple, in the following piture whih re the legs nd whih is the hypotenuse? x 10 ft. 6 ft. The hypotenuse is lwys opposite (or ross from) the right ngle nd is the longest side. So the hypotenuse in this piture is 10 ft. Tht mens tht the 6 ft nd the must e the two sides. Notie tht the legs n lso e identified y the ft tht they re the sides tht mke up the right ngle. Now sustitute into the Pythgoren Formul to solve for. +(6) =(10 36= =64 = 64 =8 So we know tht the missing side length is 8 ft. in this prtiulr tringle. Solving for Missing Hypotenuse Let s now solve for missing hypotenuse. Rememer tht the hypotenuse is lwys the longest side nd the side opposite the right ngle. Tke look t this exmple. 8 ft. x 15 ft. Note tht 8 ft. nd 15 ft. must the lengths of the legs sine they mke up the right ngle. Tht mens tht in this se is the missing hypotenuse. Plugging those vlues into the Pythgoren Formul yields the following: (8) +(15) = = Be reful t this point. Mny students mistkenly try to sutrt either 64 or 225 from oth sides, ut tht is not urte. We lwys omine like terms efore using inverse opertions, nd in this se we still need to omine the to get 289. So our next steps should look like this: 289= 289= 17= This mens tht the missing hypotenuse length is 17 feet. Note tht the only inverse opertion we needed to pply in this se ws the squre root. 291
4 Let s look t one more exmple of solving for missing hypotenuse. Consider the following piture. y 3 m. 4 m. Note tht is the hypotenuse in this se euse the sides with lengths 3 nd 4 mke up the right ngle. Plug these vlues into the Pythgoren Formul. 3 4 = 916= 25= 25= 5= So the hypotenuse hs length of 5 entimeters in this se. Pythgoren Theorem Word Prolems The use of the Pythgoren Theorem n pplied to word prolems just s esily. For exmple, if we know tht it is 90 feet from home plte to first se nd 90 feet from first se to seond se, how fr would the ther hve to throw the sell to get runner out who is steling seond se? The est tip to give for solving word prolems like this is to drw piture. In this se, note tht the distne from seond se to home plte is the hypotenuse of the tringle. Tht mens tht the 90 feet distnes re the legs. We n now solve s follows. (90 90 = = 16200= 16200= For this prolem, there ws no ext squre root. Tht mens tht is irrtionl nd it s proly est to estimte this numer. Our nswer is pproximted to the nerest one deiml ple giving us out feet. So the ther would hve to throw just over 127 feet to get out the runner trying to stel seond se. 292
5 Lesson 8.1 Find the missing side length of eh right tringle. Round your nswers to three deiml ples if neessry
6 h Solve the following prolems. Round your nswers to the nerest whole numer when neessry. 16. You re loked out of your house, nd the only open window is on the seond floor 25 feet ove the ground. There re ushes long the side of the house tht fore you to put the se of the ldder 7 feet wy from the se of the house. How long of ldder will you need to reh the window? 17. She tkes off from her house nd runs 3 miles north nd 4 miles west. Tired, she wnts to tke the shortest route k. How muh frther will she hve to run if she heds stright k to her house? 294
7 18. Televisions re dvertised y the length of their digonls. If 42 inh television mesures 18 inhes high, how wide is the television? 19. A soer field is 100 yrds y 60 yrds. How long is the digonl of the field? 20. You ple 24 foot ldder 10 feet wy from the house. The top of the ldder just rehes window on the seond floor. How high off the ground is the window? 21. A retngulr grden mesures 5 feet wide y 12 feet long. If hose osts $5 per foot, how muh would it ost to ple hose through the digonl of the grden? 22. A retngulr dog pen is 3 meters y 4 meters. If hin osts $1.75 per meter, how muh would it ost to put hin long the digonl of the pen? 23. A retngulr prk mesures 8 miles long y 6 miles wide. The prk diretor wnts to put fene long oth sides of the tril tht runs digonlly through the prk. If the fene osts $150 per mile, how muh will it ost to uy the fene? 24. A retngulr pool hs digonl of 17 yrds nd length of 15 yrds. If the pint osts $2 per yrd of overge, how muh will it ost the owner to pint the width of oth ends of the pool? 295
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