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10.3 Notes

Added Apr 01, 2015 •

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okay 10.3 last time we learned about central angles and the arcs that they made today we're going to learn about chords and the Arts that they make now remember by definition that a chord is a segment whose endpoints are on the circle  Annette Court did a circle into two arts and you have two results at follow if the arc is a diameter of course you're going to end with two semicircles  and if the court is not a diameter you're going to end up with a minor Arc  Anna major arc  so if this is a bee is your cord you're going to have baby wasp another point over here so we can name the major and Arc ATV are made okay okay now all the labels kind of got left off of these circles so if you want to Posite in label like I'm doing right here you can't but there's three rules arcs and chords in a circle in the Arts that are made by chords and we have four theorems today but three basic principles the first one is theorem 10.3 and what temp .3 says is that in the same Circle or in congruent circles two minor arcs are congruent play the arcs are congruent if and only if their corresponding chords are congruent so see we have these two Arts made by Courtney F and chords CD and what it's saying is those Arts are congruent if the cords that make them congrats are congruent if their chords are congruent okay so when this case if CD is congruent to EF so that's congruent to that then Arc CD is congruent to Arc EF so if the cords are can the Arts that they could offer congruent that's number one alright number to theorem 10.4 in 10.5 they kind of go together they're not perfect Converses of each other but they pretty much are the converse 10.4 says that if one chord is perpendicular bisector of another court so let's look at that which way should I meant if one cord is the perpendicular bisector so C here we've got a perpendicular and if that bisects that cord then the first chord is the diameter so see this is the diameter it by sexist cord and it's perpendicular okay so that makes this half to be the diameter so this would be Circle M 10.5 says if the diameter of a circle is perpendicular record so if this is perpendicular to any chord then the diameter bisects the chord and its Arc so not only is it going to buy sick that cord it's going to be check this part right here alright so if this is a b c like d right there and let's make that cake so if the diameter is perpendicular to the cord it's going to dissect the cord and park okay diameter is perpendicular to a chord it's going to bisect it and the ark that it makes right here so this Arts going to be bisected that courts going to be bisected alright so in this case we're going to have a d is going to be congruent to DB and Arc AC is going to be congruent to Arc CB all right now when it says diameter it can also be a radius but that's what's going to happen Okay the third property 10.6 Dirham in the same Circle or in congruent circles now this has circulation on it has to be the same Circle or it has to be in congruent circles two chords are congruent if and only if they are equidistant from the center if and only if means it's true both ways so how do you measure distance cuz it says their equal distance from the center remember that distance has to be measured by the perpendicular pause the sinner is right here that's M it would have to be the perpendicular line to that cord and the perpendicular line to that Court alright so if these are equal so they're the same distance from the center that means those two chords are congruent and if the chords are congruent then the arcs are congruent all right so two chords are congruent if they are the same distance from the center hey so if this is Ark Arts cord Abby and this is cord the f since those are the same distance then we know that Abby will be congruent to EF okay quick review of the three properties number one two minor arcs are congruent if their chords are congruent number to if the diameter is radicular to a cord and the art and then two chords are congruent if they are the same distance from the center alright so we have pull a little examples right here that we're going to look at so we have this circle over here in it says narc that's congruent too cute sing I got left off of this diagram is that unit do know that this is perpendicular okay so if that's perpendicular now does if a diameter is perpendicular to a court that does include a radius cuz of radius is part of a diameter soap if PQ is perpendicular 2rm then that the bisects this and it bisects The Arc so now we know that that go to that and that is equal to that name an are congruent to QR we would be cute m  I should have an arc symbol on it by the way number to if PR right here is 13 and REM this entire thing right here 24 we know it's cut in half so we know that each side is 12 that's 12 that's 12 then find the distance from p200 so we were find that right there well what do you have right here a right triangle so used to diagram think you have the hypotenuse is 13 so there M squared equals a squared plus b squared will do 12 square then you can put an X where the as that's 169 a squared + 144 subtract the 144 over here and you get 25 equals Square take the square roots 08 is 5 soapy equals 5 make it 2 p.m. so here's p.m. what is that it's from The Sinner and it goes to the circle that's a radius this is a radius also all radii there's another radius it's so every single one of these has to be 13 so p.m. would be will it to either p q  or PR and by the way if we found out this segment five could we find this one it's a radius it has to be 13 so that would be 5 this part would be 8 and you can figure that out alright the next one right here in Circle M MD is 15 m d is not drawn so you have to draw it NCD this entire distance 24 so each side is 12 find the value of x that's where the X should be alright so again you have a right triangle right here use the Pythagorean theorem 15 is are hypotenuse 12 squared plus x squared so this is - 25 equals 1/4 4 + x squared subtract the 144 and you get 81 equals x squared take the square root equals 9 number 1 through 3 right here so you can when did a number 2 but it says the measure of GH is 100 so GH that art is 100 now if these two chords are well what do you know about these two hearts well they're both going to be 100K then it says find the measure of IJ well that answers your question that's 100 okay so then we're going to start with a new set of points or not points but I use the measure of GI is 55 measure of HJ is 115 find the measure of g x now we know that they all have to add up to 360 cuz there's 360 degrees in a circle and we do know that these two have to be the same  can you can set up an algebraic problem and add all of these together and send me cool to 180 or add these two together subtract from 360 and / 2 cuz you know those have to be exactly the same amount so if you do that one 15 and 55 170 so when you subtract that from 360 you get 190 divide that by 2 and of course you get 95 so GH is 95 and IJ would be 95 also so these are 95 degrees okay now number 3 I'm going to do an color IJ is 85 so this is 85 + h j  h2j is 120 and you need to find GI right here well if that's 85 this is 85 right so add 85 120 and B5 together and you get $209 subtract that from 360 and you get 70 so that is 70 degrees add up to 180 all right now let's look at number for number 4 is just a little bit different it's one of those that you would people use in real life to be able to I've learned a couple of different ways to do this here's a whole nother way to do this three cities are located on a map as shown  right here a wireless Communication Company plans to build a relay Tower to provide cell service to each of the cities in order to provide the city the cities with a signal of equal strength never must be placed equidistant from each show and explain how this location can be determined so we want to find where to put this cell tower that says same distance from each one of the city's now if you think about a circle  the radius is the same distance so if we can find the center of the Circle that is made by these three then we can find the center of all of them now the way that you do this is number one you're going to connect the cities like the skank then number two if we can find the perpendicular bisectors the radius or diameter has to be the perpendicular bisectors of these chords that are made by the circle so think about the circle and if I do a perpendicular bisector they have to go through the center of the service so what you're going to do is you're going to draw of course you would do this technically with the protractor and ruler but find the center of this one and you're going to draw a did killer bisector that goes through there and cuts it in half then you're going to do the same thing with this one that's off a little bit but let's go with it so right here where they cross at what should be a little lower is going to be the same distance to each City so this would be the point that you would place the cell how are to get equal distance to every city think about it in a circle so here's your two chords right here here is city a b c the center is here so these would be the radius would be the same to each distance for each City in that case and remember that if I draw the perpendicular by after that's that in half cuts that in half  I found the center of the circle so this is a Circle image of what you're doing right here one more problem and we're done in circle a find the links of CJ and a keg in BC equals f g equals 16 and AJ equals 5 now this is a problem separate from number five and six Okay so this one BC  an FG or both 16 so FG and VCR both 16 will that's perpendicular that's condition and it's coming from the center Suite cut it in half half a 16 is 8 so this would be eight this one would be eight this one would be eight and that eight would go for kids me and CJ so then it says AJ is 5 so what is the distance from 2J to fund the first one well see you today right there would be 8  so CJ equals 8  nak would equal right here remember if these chords are equal then these are the same distance from the center so if that one's 5 this one spot so AK would equal 5 and that's that problem now when we start on them lowers different sings here but we do know that these cords are equal so it says that Arc CD is more X  an arc BD is 2x + 40 so if this line is perpendicular which it's shown it bisects this Arc and it by 6 is court so we know that these two have to be equal so we would just simply do for x equals 2x 40 subtracted 2 x 2 x equals 40 x equals 20 so says find the measure of Arc BD so you need to plug it back in since or equal you can plug it back into either one so if we plug it back into the 4 x 4 * 20 is 80 degrees so that's that answer all right and last one number 6 find gkg okay right here if BC is 20 nfg is 20 well if f g is 20 what is GK half a 20 would be Tim and you are done say your homework is a worksheet I'll give it to you in class tomorrow if you'll just have the notes field in and we're going to go over him and go over some other stuff during class don't forget that page 655 and 661 are due when you get to class from the last section

Uploaded by Pam Chastain.

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10.3 Notes

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