Notes 10.6 Segment Lengths in Circles

[00:00:00]hey guys and welcome to notes 10.6 on the segment lengths in circles so to start us off we're going to talk about two intersecting chords if two chords intersect then the product of the segments of one chord is equal to the product of the segments on the other chord so in my picture right here i have a pink chord and i have a green chord all right so the pink chord what i can say about that is that if a e this piece right there a e times the other part of it which is e b ae times eb should be equal to the green chords split as well which is d e times e c those two products shouldn't be equal since these two points are intersecting let's see this in action look at our first example down here we have a lot of numbers we've got 10 15 18 then of course we have our x so again this is just understanding the property from up above we have this piece of it right here and that piece of it right there these are two chords intersecting so i can do 15 times x and it should be equal to the other part of it i have the 10 times the 18.

[00:01:11]now this is probably easiest done uh just straight up doing the calculations we get 15x here on the other side we get 180 and as you solve for x of course the only thing you want to finish up by doing is divide by 15 you quickly figure out the 180 divided by 15 gives you x is 12.

[00:01:29]you'll find an example to be very similar to this you guys can do that one for your quick check let's keep going let's look at example number two i want to find the value of x in this problem and we have a little bit more algebra involved now so in this problem we still have the same overall principle here we have the 12 and the 3x plus 1 intersecting the other chord 8 and 24.

[00:01:48]so we can go ahead and say 12 times 3x plus 1 needs to be equal to the other chord is eight and twenty-four so eight times twenty-four all right well you do need to distribute on the left there and twelve times three gives us 36x and twelve times one gives you twelve and on the right half 8 times 24 gives you 192.

[00:02:11]as you move in to solve this we do want to go ahead and subtract 12 from both sides trying to isolate my x as i subtract i get 36x is equal to 180 and now all i need to do is divide both sides by 36 as i divide both sides by 36 we find out that x is equal to 5. you'll find example b to be very similar to example a please use mine as a reference let's look at our next rule down here we have two intersecting secants so this rule says that if two secants intersect then the whole secant times the outside part of the first secant is equal to the whole secant times the outside part of the second secant now that's kind of a mouthful which is why i didn't put any fill in the blank there but i want to go ahead and show you what that means i think it's actually best understood with the diagram down here in color so you see i kind of have the pink parts or the whole pieces and hopefully you can see the green parts there those are the partial pieces right and so what we want to understand here is that if you do one side for example if you do like ac times ba or a b however you want to write that right ac times a b i'm completely doing the top piece here it needs to be equal to the other side done the same manner so the other side is a e for the whole piece and the other side is a d for the partial piece the outside piece so the pink representing the hole in the green representing the outside part of the sequence all right and so this is our rule and again we could rewrite this up top here because it is the same property just without color ac times a b is equal to a e times a d the more you practice it the more this will become second nature to you and you'll just kind of just know let's look at an example and see this in action so for example number three i want to find the value of x a lot going on here i've got an x a 12 a 15 and 25 but i understand that follow the same rule that we just talked about so we're going to go ahead and take the outside piece times the whole piece on one specific side so maybe you want to look at just the top part here all right on just the top part there we can go ahead and say that the x which is the whole piece times the 12 which is the outside piece if we follow the same pattern on the other side these two things should be equal so what is the whole piece on the other side well be careful a whole piece on the other side you actually have to add 15 plus 25 to get the whole thing which of course is going to be 40. but be very careful because sometimes that'll be the trick is that you actually need to add to get the whole piece there we do have 40 times now just the outer piece just like 12 there the outside piece is 15. and so this is how we're going to solve this now of course there's not much to do here 12 times x is 12x and 40 times 15 that does give you a pretty large number there 40 times 15 will get us to be up to 600 and then if we just go ahead and divide both sides by this 12.

[00:04:57]600 divided by 12 will get us down to 50.

[00:05:00]and that's what our x value would be the whole distance of that top side would be 50.

[00:05:05]if you look at example b it follows a similar manner to the one i've just worked here just be very careful make sure you're not following any traps let's look at another example in number four here i want to find the value of x now this one's a little bit different okay we want to make sure we get the whole side with the outside in both scenarios all right so let's look at just one side first let's look at this side of the x the whole side there the whole side is x plus five so we're gonna have x plus five being the whole side we're gonna multiply that by just the outside piece what's the outside piece it's a five so if x plus 5 times 5 follow that pattern on the other side all right the whole thing on the other side 11 plus 9 gives us 20. so the whole thing is 20 times just the outside piece of course is 9.

[00:05:50]so this one we have a little bit more to do here we actually have to use our distributive property so we're going to go ahead and multiply inward here we have 5 times x give us 5x and 5 times 5 25. on the other side we have 20 times 9 which is 180 and now we're down to just solving for x let's go ahead and subtract 25 from both sides 180 minus 25 will be at 155 and divide by 5 on both sides as we do that find out x is equal to 31. so again just be very careful make sure you're aware of what the whole side is because the whole piece times the outside piece that's what we're looking for here if you look at example b it follows a similar manner uh use mine as a reference let's look at the last thing that we're going to have in this chapter which is one secant and one tangent all right so it says if a secant and a tangent intersect then the whole secant times the outside part is equal to the whole tangent times the outside part it's actually pretty much identical to the property we had before this just mixes one secant one tangent so in this part here let's just look at the secant part first okay we take the whole secant all right the whole secant is a c if you multiply by just the outside part that's just a b just like before right now tangent very very similar all right and the tangent has the a d being the outside part and then in this one the outside part of the tangent is also actually a d so it kind of just repeats itself there when we had the tangent part of it okay let's see this one in action in a couple of examples see what happens here it's gonna be a little bit different it says find the value of x and we notice we have a secant and a tangent so let's start with the secant part the whole secant from end to end six plus twenty six plus twenty is twenty six take the whole secant times just the outside part of the secant which is six so 26 times six is equal to now the other part the whole tangent is x and the outside part of the tangent is also x so we can have a repeated thing there just like we did in our in our example with the notes so 26 times 6 that will make um that'll make 156 for you and then x times x hopefully you know that one x times x is x squared and to figure out what x is we're going to actually have to go ahead and take our square root of both sides here as you take the square root of 156 all right the square root of 156 you will have to use a calculator and when you do you can round to the nearest tenth which is 12.5 always follow the directions i think most of them do sit around to their tenth but make sure you read the directions and just double check and there's your answer so let's look at one last example here b is found in your quick check just follow example a and you'll be all right look at our last one together here example number six i'm gonna find the value of x all right so in a similar manner here remember you always need to figure out the whole secant and the the outside secant so what's the whole secant all the way across here well the whole secant is 3 plus x just the outside part that's the 3. so we're going to take the whole secant 3 plus x times the outside part which is 3.

[00:08:58]make that equal to the outside part of the tangent sorry the whole the whole tangent there times the outside part of the tangent it gets a little confusing because it's the same thing right so 8 times 8. so hopefully on the left you notice you do need to distribute 3 times 3 and 3 times x 2 times 3 is 9 3 times x is 3x on the right 8 times 8 gives us 64.

[00:09:18]let's go ahead and solve this we're going to go ahead and subtract 9. 64 minus 9 will put us at 55 and as we divide by 3 on both sides to figure out what x is we are going to get a decimal again you can always round to the nearest 10th if necessary but this time we get 18.3 as you move in to do the last example example b it is found in your quick check and workspace limits one i've just worked here so please use mine as guide you guys have any other questions don't be afraid to reach out