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10.3 notes AP

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10.3 is all about chords in circles so the last one we talked about angles the central angle specifically made by two radii and now we're going to talk about chords and remember a chord is a segment  line segment specifically that has the endpoints on the no court is going to divide a circle into two arts and it's going to have to situations if the court is a diameter 7 - chord is a diameter it's going to cut the circle in half so you're going to get a semi circle  and you're going to get two of them K know if the chords not a diameter here's our cord gets going to cut off to Art it's going to cut off this Arc it's going to cut off this art now this is just like our central angle this is going to be a minor Arc  and it's going to last 180° and it's going to cut off a major arc and that is going to be in art that is greater than 180° so when you have a cord your hat one of these two situations you're going to get two semicircles if it's the diameter or any other cord is going to be a minor Arc and a major arc rules that we're going to learn about these cords in the Arts that they make in a circle the first one is 10.6 and it said in the same Circle or congruent circles I remember same Circle congruent circles that have the same radii two minor arcs are congruent if and only if their course chords are congruent so what that saying is if you have two chords in the same Circle if the cords are equal in distance then the arcs are going to be in their measurement so two minor arcs are congruent if the chords are congruent can I serve two cords of congruent the minor arcs they make are congruent our next one is serum 10 seven and and I put these together because they're just the converse of each other okay so 10 7 says if a diameter of a circle is perpendicular to the court so diameter is perpendicular to a chord then the diameter bisects the chord and its Arc so what you're going to have here is that this is cut in half so F H and H deer the same and the arc that minor Arc it makes is also cut in half so art FG ngd are going to be the same now 10.8 is just the converse of that if one cord is purple is the perpendicular bisector of another chord then the first chord is the diameter so here I have two courts if one of them is the perpendicular bisector then that has to be the Diane search two and then it bisects and is perpendicular to the other court so in other words let's wrap this up if the diameter is purple killer to the court  it bisects the cord The Arc so in this picture specifically FH is congruent to HD and Arc FG is congruent to GD okay before we go to the third one let's just go over this one more time if two chords are equal then the arcs are equal if a diameter is perpendicular record it cuts the cord in half and The Arc in half what four number 310 9 in the same Circle or congruent two chords are congruent if and only if means it's true busbice they are equal distance from the center so remember we measure what's a perpendicular line so if CD and a b are the same distance from the center that means that eat a g and e deaf would have to be equal in CD and ab are congruent to two chords are congruent if they are the same distance  to the center so here if BG and E F are equal then CD is congruent to ad and think about  one of our other things we just learned that first one then if the chords are congruent then we know the arcs are congruent so they build on open each other let's do some examples  okay in this example we're going to use this circle name an arc congruent to Q heart  so if we have QR right here you notice that this is perpendicular to the court so QR is congruent to qm that heart okay now on this one if PR is 13 arm this whole thing all the way across is 24 can you find po now since perpendicular we know what cuts this and hats with the whole thing is 24 then this is 12 and 12 can you find po that distance well because that we know that this is perpendicular that makes two right triangles so we can use Pythagorean theorem if you notice that's a triple you can do that but if you don't you're going to have to use Pythagorean theorem which side is c one across from the 980 so this is 90° right here so this across here is going to be your seat so 13 squared equals 12 squared plus X squid that's 169 equals 144 plus x squared subtract the 144 and you get 25 equals x squared take the square root x equals 5 so po equals 5 so we know that that's five now alright then it says right or name a segment congruent to p.m. so let's leave that p.m. how do we know what's can run to that well what kind of segment is that it starts in the center and goes to the circle that's a radii will show his PR so if PRS 13 p.m. 13 so PR is congruent and they both equal 13 in Circle M is it till the same as the center so this line right here if we extended it will be our radii MD is 15 CD is 24 find the value of x is very similar to the problem we just did however MD is not drawn here but what is MD that starts at 2 goes to a point on the circle that's a radio so we know this is 15 if you want to you can go ahead and draw this radio it's 15 every radius you draw in the circle 15 and then we know that CD the whole thing across is 24 now if that's perpendicular cuts the cord in half for this is twelve and this is twelve to find X we just use this right triangle and use Pythagorean theorem here's our 90° all right here so this side is RC so we would do 15 squared equals 12 squared plus x squared so that 25 equals 144 plus x squared subtract the 144  how long did 81 equals x squared take the square root of both sides x equals 9 so X is 9 okay now let's put what together and do these examples no remember your three things if the cords are equal the Archer equal if the diameter or radius is perpendicular to a chord it cuts Gordon half and The Arc in half and if two chords are the same distance from the center than they are equal to each other and if they're equal their arcs are equal look at this problem let's make it bigger it says if g h is 100 fine I J Will since these two or equal their archery equal so that's really easy it's also a hundred degrees okay so let's start with the next problem now each one of these is a new problem so we're not going to use this hundred again if GI is 85° HJ his 115 can you find GH now think about that we know that from here and here are the same so we can name them both ex now these all have to add up to how much 360° so I could make an algebraic problem out of it and do x + x + 115 + 55 360 or you can just do the math and add these two together should drive from 360 and then we know these have to be equal so divide what you have left by 2 so 55 + 115 is 170 so subtract that from 3 turn 60 can you get 190 now divide the 190 by 2 get 95 here and 95 here so GH is 95 degrees okay now for the problem we're going to have to cross this out and use new stuff okay let's do this when I'm Billy alright I J which is turn up here is 85° HJ here is 120 degrees fine GI well you know that this Argan the sarkar the same soap this is 85 this one is 85 so at51 20 + 85 + what they have to add up to 360 so add those together can you get to 9360 - 290 you get 70 so GI is 70°  okay this one read your change in label everything before we can do these two problems okay and circle I find the length of CJ and AK given that B C and F G both equals 6 so let's look at that BC MFG both equals 16 is perpendicular so the 16 is cut in half so this is eight and this whole thing is eight same thing here in this is 8 and then your day is well as that's being 8K then it says AJ is 5 okay well think about that if this is six turn this is 16 Seas chords are equal so they're the same distance from the center so if that is 5 distance this has to be five so AKs five we found that  and CJ is the other thing we have to find which look at CJ well that's perpendicular so it cut it in half so CJ is 8 so we found both of all right now let's look at number 5  okay no read this one real carefully we're going to find the arc V D that's this one okay if CD is 4X and BD is 2x + 40 know what do you know about these two are if this is perpendicular that cord it cuts the cord in half Starkey in DC for Egypt so these two things are going to be equal to each other so we're going to do for x equals to + 40 subtract the two x over here to x equals 40 divided by 2x equals 20 now we go find BD to plug that back into your ex 2 * 20 + 40 so that is 40 + 40 80 degrees soubd equals 80 degrees k-6 find  GK now let's let GK is half of this right if BC nfg equal 20 so these are both 20 what is GK have to equal well 20 they're equal that bisects because it's perpendicular so half a 20 is 10 so your answer would be 10 for G okay right here and then number 7 let's look at it and eight fun value of x here look what we have we have a diameter we have a cord they are perpendicular if the diameter is perpendicular to a chord it cuts the cord in half and The Arc in half so this amount will be equal to this amount because that's for that part and that's for that part it's all you have to do is 3x 5 equals 2x + 1 subtract the two x over here had the five over here x equals 6 when you're done K and number 8 look at this one  find X this court is 20 this court is 20 if the cords are equal and what do we know about their distance to the center their distance to the center is saying so this has to equal that so for x - 5 equals x + 4 subtract your ex here add your five here you get 3x equals 9 / 3x equals 3 2 x equals 3 now that has to be in the same Circle or in what circles meaning they have the same radius if a diameter or radius is perpendicular to a chord then it cuts the cord in half and the Ark and if two chords are equal they're the same distance perpendicularly from the center

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