Notes The Pythagorean Theorem, Distance Formula, Midpoint Formula

Transcription

1 Notes The Pythgoren Theorem, Distne Formul, Midpoint Formul The Pythgoren Theorem CCSS 8.G.: Explin proof of the Pythgoren Theorem nd its onverse. In right tringle, the side opposite the right ngle is lled the hypotenuse. The legs re the sides tht form the right ngle. We typilly lel the legs nd, while the hypotenuse is. hypotenuse legs If we look t enough right tringles nd experiment little, we will notie reltionship developing. Drw right tringle on piee of grph pper, with leg mesurements of 3 units nd units. Drw squre on eh side of the tringle s shown. Cut out squres nd nd ple them side y side. Tpe the repositioned tringles. Compre your new squre with squre C. C Drw two tringles s shown; ut long the hypotenuses nd slide s shown. It ppers tht the sum of the res of the squres formed y the legs is equl to the re of the squre formed y the hypotenuse. There is nother wy to look t this reltionship. Grde 8: The Pythgoren Theorem, Distne Formul, Midpoint Formul Pge 1 of 9

2 egin with right tringle, legs nd with hypotenuse. Crete squre tht hs mesure + on side, using ongruent tringles s shown. The re of this new squre n e expressed two wys: 1) = ( + ) (length of side is +, so multiply sides to get re) 1 ) = + ( tringles, eh with re of 1 ; dd to the re of the squre with side ) Setting these two vlues equl to eh other, we get ( ) + = = + = + = This is n importnt reltionship in mthemtis. Sine it is importnt, we will give this reltionship nme, the Pythgoren Theorem. nd yet nother wy to model the theorem: Grde 8: The Pythgoren Theorem, Distne Formul, Midpoint Formul Pge of 9

3 Drw right tringle on piee of grph pper, with leg mesurements of 3 units nd units. Drw squre on eh side of the tringle s shown. Cut out squres eh, ongruent to nd, in two different olors Glue one nd one on the originl grph. Lel with the re. C re = 1 units re = 9 units C Hve students ut nd pste the remining squres (9 + 1) to form third squre C, whih represents the re of 5. Hve students ut nd pste the remining squres (9 + 1) to form third squre C, whih represents the re of 5. re = 1 units re = 9 units re = 5 units Finlly, hve students lel the legs of the originl right tringle with nd, hypotenuse. (This represents one possile solution.) It ppers tht the sum of the res of the squres formed y the legs is equl to the re of the squre formed y the hypotenuse, or + =. Pythgoren Theorem: If tringle is right tringle, then + =. While these models re helpful, the stndrd does sy to explin proof of the theorem. For more disussion regrding this, plese go to the following doument: The onverse of the Pythgoren Theorem sys If tringle. Let s see if this seems to e true. + =, then the tringle is right Grde 8: The Pythgoren Theorem, Distne Formul, Midpoint Formul Pge 3 of 9

4 Fill in the first empty olumn in the hrt elow. Then, use grid strips ut to one squre width to rete the tringle with the 3 side lengths given (see exmple elow the hrt). Does it pper to e right tringle? Is + = true? Does this pper to e right tringle? = 5 true Yes, see elow = Looks like right tringle. To model the onverse, use the exmple of the Egyptin rope-strethers. Some nient toms depit sries rrying ropes tied with 1 eqully sped knots. It turns out tht if the rope ws pegged to the ground in the dimensions of 3--5, right tringle would emerge. This enled the nient Egyptin uilders to ly the foundtions for their uildings urtely. To model this, deide on unit length nd mke 13 knots (representing this unit s mesure) in piee of rope or twine. This wy you n hold the first nd lst knot together. Using 3 people you n streth the rope into 3--5 tringle nd model where/how the right ngle ourred. CCSS 8.G.7: pply the Pythgoren Theorem to determine unknown side lengths in right tringles in rel-world nd mthemtil prolems in two nd three dimensions. Exmple: Find the unknown length in simplest form. + = = = 19 = 19 = 13 = 5 1 Plese note tht this is one of our Pythgoren triples. Pythgoren triple is set of three positive integers,,, nd suh tht + =. nother triple worth noting is Grde 8: The Pythgoren Theorem, Distne Formul, Midpoint Formul Pge of 9

5 You n rete other Pythgoren triples y multiplying the originl triple y ftor. For instne, 3--5 eomes -8-1 y multiplying ll the sides y. One might rete nother, 1--, y multiplying the y. This skill is very vlule when working with prolems on the Nevd CRT nd NV High Shool Profiieny Exm. It n sve the student time nd frustrtion! Exmple: Find the unknown length If we reognize the ommon ftor of 5, we n think: This is leg nd the hypotenuse of the So the missing side would e 5, whih is. We hd 3--5 tht ws multiplied y ftor of 5 to produe So =. Exmple: 1-foot ldder is pled ginst uilding. The foot of the ldder is 5 feet from the se of the uilding. How high up the side of the uilding does the ldder reh? Give your nswer to the nerest foot. + = 1 < 119 < = 1 1 < 119 < 11 1 ft 5 + = 1 1 < 119 < 11 = 119 Sine 119 is loser to 11, 119 is loser to 11. = 119 Therefore, the ldder rehes pproximtely 5 ft 11 feet up the side of the uilding. Exmple: You use four stkes nd string to mrk the foundtion of house. You wnt to mke sure the foundtion is retngulr. The four sides mesure 3 ft, 7 ft, 3 ft nd 7 ft. You then mesure digonl to e 78 ft. Is your foundtion retngulr? The digonl divides the foundtion into tringles. If the tringles re right tringles, + = will e true sttement. euse = 78 ( = 8), you n onlude tht the tringles re right nd your foundtion is retngulr. Grde 8: The Pythgoren Theorem, Distne Formul, Midpoint Formul Pge 5 of 9

6 Exmple: Find the length of the longest line segment (lled the min digonl) in the retngulr prism shown on the piture elow. Solution: We will pply the Pythgoren Theorem twie. First, lel the points nd sides tht will used on the piture. We will find x using the Pythgoren Theorem in tringle C. Then we n find y using the Pythgoren Theorem in tringle CD. For C, + = + 5 = x = x 1 = x We do not hve to simplify x s we need it in tht form in the Pythgoren Theorem: For CD, + = + 1 = y 5 = y 5 = y 5 = y The length of the min digonl would e 5 feet (or pproximtely.7 feet). Note: Our result is tully Indeed, we n see tht the length of the min digonl in retngulr prism with sides x, y, nd z is L= x + y + z. This is sometimes lled the 3-dimensionl Pythgoren Theorem. Grde 8: The Pythgoren Theorem, Distne Formul, Midpoint Formul Pge of 9

7 nother Retngulr ox prolem: You hve huge retngulr ox with dimensions1ft 15ft 9ft. Wht is the length of the longest rod you ould fit in your ox? 9 ft 15 ft 1 ft Solution: We will pply the Pythgoren Theorem twie. First, lel the points nd sides tht will used on the piture. We will find C using the Pythgoren Theorem in tringle C. n find D using the Pythgoren Theorem in tringle CD. Then we For C, = x D = x 39 = x For CD, = y 5 = y 9 ft 1 ft 15 ft C 5 = y 5 = y The length of the longest rod would e 5 feet (or pproximtely 1. feet). The Distne nd Midpoint Formuls CCSS 8.G.8: pply the Pythgoren Theorem to find the distne etween two points in oordinte system. Tke the time to link the distne formul with the Pythgoren Theorem. Here s wy to show it. You re given two points, (3, 1) nd (, 5). You re sked to find the distne etween them. Plot the points. You n drw right tringle, using these points s two of the verties, s shown. Grde 8: The Pythgoren Theorem, Distne Formul, Midpoint Formul Pge 7 of 9

8 Find the lengths of the horizontl nd vertil sides of the right tringle y sutrting the x- nd y- vlues. or You ould simply ount the units from one point to the next. (3, 1) (, 5) 5 1= We now will use the Pythgoren Theorem to find the missing length, whih is the hypotenuse. = + ( 3) ( 5 1) = + = 3 + = = 5 = 5 = ± 5; however, sine we re looking t mesurement, only the positive vlue will e onsidered. 3= 3 Now, insted of using the ordered pirs (3, 1) nd (, 5), we use pir of points, ( x1, y1) nd ( x, y ). Sustituting into the Pythgoren Theorem one gin: = + ( ) ( ) = x x1 + y + y1 ( ) ( ) 1 1 = x x + y + y Sine we re determining distne, we reple the with d, nd we will only onsider the positive vlue. So we now hve the distne formul: ( ) ( ) d = x x + y y 1 1 Exmple: Find the distne etween the points (, 1 ) nd (, 5) ( ) ( ) d = x x + y y 1 1 ( ) 5 ( 1) d = +. Grde 8: The Pythgoren Theorem, Distne Formul, Midpoint Formul Pge 8 of 9

9 ( ) d = + d = d = ( 5 is etween 7 nd 8, ut loser to 7) d = 13 d = 13 The distne etween the points (, 1 ) nd (, 5) is 5 or 13 units. Sine 9 < 5 < 7 < 5 < 8 3 nd. then 5 7. units 15 = Students need to know the midpoint formul lso. Consider the line segment with endpoints (, 1 ) nd (, 5 ). The midpoint is the point on the segment tht is equidistnt from the endpoints. We re looking for the middle! If we think of this s finding the verge, we ould dd the two x vlues nd divide y. We ould likewise verge the y vlues. Finding the verge for our exmple, it ppers the midpoint is t (, 3). x1 y1 x y, we n rrive t the formul for finding the midpoint: x 1+ x 1, y + M y = If we use endpoints (, ) nd (, ) Exmple: Find the midpoint of the segment with endpoints ( 5, 7 ) nd ( 1, 5). x1+ x y1+ y M =, 5+ ( 1) 7+ ( 5) M =, 5+ 1 M = (, ) The midpoint of the segment with endpoints ( 5, 7 ) nd ( 1, 5) is (, ). + Grde 8: The Pythgoren Theorem, Distne Formul, Midpoint Formul Pge 9 of 9