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10.6 Notes AP

Added Apr 14, 2016 •

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10.6 finding segments links now we've learned how to find all the angles that can exist in and outside of a circle now we need to find out how to turn the links of the segments that are inside a circle so quick review look at this name each segment or line in circles now before we start we need to make a correction Mark out that be and make it an F okay so no look over here line maybe what is this called from A to B diameter  okay what is this called line from C to D what is that called radius  and then line segment from h2g h2g what is that called in points are on the circle chord and then the next one is a lie not a line segment BF so EF the whole line not just the line segment EF but the whole line EF that is a secant and the line g j it only intersects the circle in one spot right there that makes it a tangent and then line segment from e2f not the line but from EDF that line segment that is a court all right you can use special segments in circles to find the measure of the segments and there's going to be a couple of theorems formulas to tell us how to do that  segments of cords now segments of chords is when two chords intersect each cord is the two segments of this segment is cutting to we have segments a and segment be this court has been cut into two pieces segment X segment why that's what it's talking about okay when we have two quarts this is there in 10.18 and what it says basically is when you have two chords that intersect in a circle then there's egg is Multiplied equal each other so X Sons why these two parts multiplied together equal the other segment Parts multiplied 8 tons be check this part times this part equals this part times this part okay so you put these two segments together on the same cord you put these two segments together on the same court so if we look at this problem right here  this segment is broken up into x + x + 4 so it'd be x * x + 4 equals the other segments this segment times each other now that's x + 1 * x + 2 so that would be X+ 1 * x + 2 from there on out it's algebra you multiply those two together equals those two okay so let's do the algebra x x x is x squared x * 4 is 4 x that's distributive property now I have to do binomials that means to an h and you've got a full it out if you want to use the box method I think you need to get used to not using the box method it just think about it in the distributive property multiply this x by both of those and then multiply this one by both of them so x x x is x square x times 2 is 2 X now the one tons each other one x axis x 1 x 2 is 2 now on this go ahead and add  your ex's together so that's 3 x + 2 - x + 1 x 3 x equals x squared plus Forex now when I go to subtract the sex Square to get it on this site I subtract the x squared from both sides what happens where my SEC squared 0 play my sex crazy records are going to cancel out so now I have 4 x equals 3x plus to move this over to this side so all the EX's are on one side and now I have x equals  too and you're done now if you go back up turn think about that that means this one is too 2 + 4 is 6 2 plus 2 is + 2 + 1 is 3 does that work 2 times 6 is twelve 3 * 4 is 12 so works out okay what if these lines go outside the circle and cross not cross inside let's we need to name all of the parts though and that's what says up your tangent segments and secant segments now these are segments not lines a tangent segment is a segment that is tangent to a circle at its in point so see the tangent Circle the tangent segment it ends where it intersects The Circle at Sops is a tangent a secant segment is a segment that contains a chord of the circle and has exactly one endpoint outside the circle this is secant segment prpr is a secant segment so all way from P it's on the outside to our but it ends on the other side of the circle the part of the secant segment that is outside the circle is called the external segment so just Pita Q is the external segman of the secant segment where they use these parts to set up this formula the next one so let's look here if we have to seek see they both go through the Circle K it has a formula in theorem 9 is 1019 if you read it it'll basically come down to this baby the external segment X AC the secant segment so the outside times the whole thing equals ad the external segment x 80 the secant segment so the outside times the whole thing so what you're going to do who is you're going to get a tab e x a to C equals 8 d x a2e okay so let's look at our example right here using this formula alright are external is for what time's the whole thing is 6 so it's 4 x 6 you have to add those two parts together to get the whole thing equals the external part is three times the whole thing of the whole thing is 3 + x and we can't add those cuz they're not like terms so it 3 and the 3 + x k the outside four times the whole thing six equals the outs what three times the whole thing three plus X now you just do Algebra that's 24 equals 3 times three is nine three times is 3x subtract the nine over here and you're going to get 15 equals 3 next / 3x equals 5K all right so remember the outside times the whole thing the outside what time is the whole thing all right what if it's a tangent and a secant okay so now we have an external segment secant segment and a tangent segment now this is very similar the top is going to be the same as the one above but it's the tangent segment that's different sister and 10.20 you can read it in your book what it says is XY from here to hear X XC the whole thing so that's turn has not changed its the outside times the whole thing now here's where it's different equals x q squared so if there's one part you multiply it by itself so you're just going to take this part and square it okay so let's look at this example out here okay no the top is this the external plus the whole thing so the whole thing is 16 + x so you'll do 16 x 6 10 + 6  and then on the bottom this 24 there's only one part so you square it and then from sir it's just your algebra 16 * 16 is 25616 * X is 16 x do not forget to distribute 24 squared is 576 okay then from there subtract the 256 and you get 16 x equals 320 divided by 16 + x equals 20 so this what's right here has to be 20  okay so for a quick recap we have three situations if they cross on the in you do this part times this part equals this part times this part so x times y equals 8 x be if it's to sequence you do the out what time does the whole thing equals the outside times the whole thing it's a tangent and a secant you do the outside tons of whole thing equals the tan segment squared that true three formulas you need to have them memorized okay let's do a couple of examples 1 through 3 find the value of x okay this one intersex inside so you're going to do 15 x 5 x + 2 equals the other 10 * 7X now just do the algebra 15 x 5 x 75 x 15 * -2 is -30 equals 10 * 7x is 70 X  then just because this side only has an X right now I would move the 75 over here deal with the neg so I would subtract the 75 x on this side and get -30 equals -5x now if you divided by -5x equals 6 positive 6 negative divided by negative is positive so in this one you could go back and check and plug in your 6 here and here and make sure that multiplies and that multiplies to the same number number two on this one I have a secant segment and a tangent segment so I'm going to do the outside time's the whole thing equals the tangent squared now before you start what is the whole thing here so I'm going to do this one time what's the whole thing will the whole thing is x - 1 + 5 so that is -1 plus 5 is 4 so it's X for when you add them together okay so the outside is going to be x -1 x the whole thing is X Plus more now  the tangent segment right here is squared so that equals x + 1 Square now be careful because you've got to remember what that means that means x + 1 * x + and that's what you're actually going to write out so we're going to have to fall out both of them or distributive property if you want to think of that term do you do the xmcx in the X X the 4X x x is x squared x x 4 is 4X now do the negative one tons is -1x and the -1 * 4 is -4 and then we've got to do the same thing here so x times x is X squared  move it over so I can see it  x * 1 is 1 X or x 1 x x is 1 xrx and 1 * 1 is what now collect like terms and saw off here on this side here's my two is so x squared plus 3x 4 -1 -4 and then I can add the two x's here at square + 2 next plus one now what's going to happen to your x squared If I subtract x squared here I have to do it on this side and x squared minus x squared zero squared minus x squared is zero so now I have 3 x - 4 equals 2x + 1 now I've showed you several times how to do this in one step let's move our exes to the left so I move the 2X and put it over here negative and then I want to get rid of the -4 so go ahead and do that stuff if its -4 on this it becomes plus 4 not 5 on this site okay so that's just x equals 5 and you are done so if you want to go back up here you would have five- one is for here and 5 + 1 is 6 here so for we got to put the five here that would make that one nine so think about 4 x 936 6 squared is 36 so see it does work out okay and the last one number three now these are two secrets so it's going to be the outside times the whole thing will 3 plus 1 is 4 and then this one it's going to be the outside is 2 times the whole thing well if I add those two together that's x + 3 + 2 so that's x + 5 for the whole thing okay so let's do this 1/3 times 4 equals 2 * x + 5 - x x + 5 little bit easier algebraic problem to solve 3 * 4 2 * x 2 x 2 x 10 is 10 subtract 10 from both sides 2x equals 2 2 x equals 1 so if you go back up here and put a one here that makes the whole thing or let's put the one right here on + 5 is 6 so if we check it 3 times for 12 2 * 6 is 12 and they are all correct okay so don't forget you have three different setups for this one make sure Yuma

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10.6 Notes AP

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