Definicja 1
Okręgiem o środku O i promieniu r, r> 0,
nazywamy zbiór wszystkich punktów płaszczyzny, których odległość od punktu O
jest równa r. Taki okrąg oznaczamy o(O,r).
Promień okręgu to odcinek ( i jego długość) łączący środek
tego okręgu z dowolnym punktem tego okręgu. Okrąg o promieniu r ma długość 2πr.
Łuk okręgu to część okręgu wyznaczona przez dwa punkty
okręgu wraz z tymi punktami. Zwróćmy uwagę, że dwa punkty wyznaczają
jednocześnie dwa łuki.
Cięciwa okręgu to odcinek, który łączy dwa dowolne punkty
okręgu . Cięciwa, która przechodzi przez środek okręgu jest jego średnicą.
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Twierdzenie 1
Jeśli
promień jest prostopadły do cięciwy, to dzieli tę cięciwę na dwa odcinki mające
tę samą długość.
Odległość środka okręgu od cięciwy to długość odcinka
łączącego środek okręgu ze środkiem cięciwy.
Wzajemne położenie prostej i okręgu:- Prosta ma tylko jeden punkt wspólny z okręgiem.
- Prosta ma dwa punkty wspólne z okręgiem.
- Prosta nie ma punktów wspólnych z okręgiem.
Definicja 2
Prostą, która ma tylko jeden punkt wspólny z
okręgiem, nazywamy styczną do okręgu w tym punkcie ( zwanym punktem styczności
prostej i okręgu).
Twierdzenie 2
Prosta jest styczną do okręgu wtedy i tylko
wtedy, gdy promień poprowadzony do punktu wspólnego prostej i okręgu jest
prostopadły do prostej.
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![](data:image/png;base64,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)
Z punktu znajdującego się na okręgu można poprowadzić tylko
jedną styczną do okręgu. Z punktu oddalonego od okręgu o ponad długość
promienia można poprowadzić dwie styczne do okręgu. Jeśli odległość punktu
jest mniejsza niż długość promienia to
nie da się poprowadzić żadnej stycznej.
Twierdzenie 3
Prosta jest styczną do okręgu wtedy i tylko
wtedy, gdy odległość środka okręgu od tej prostej jest równa promieniowi.
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Twierdzenie 4 (o odcinkach stycznych)
Odcinki dwóch stycznych poprowadzonych do okręgu z punktu,
którego odległość od środka okręgu jest większa niż promień – wyznaczone
przez ten punkt i odpowiednie punkty
styczności – mają tą samą długość.
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Definicja 3
Sieczną okręgu nazywamy prostą, która ma dwa
punkty wspólne z danym okręgiem.
Twierdzenie 5
Prosta jest sieczną okręgu wtedy i tylko wtedy,
gdy odległość środka okręgu od tej prostej jest mniejsza od promienia okręgu.
![](data:image/png;base64,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)
Mówimy, że prosta jest rozłączna z okręgiem, kiedy nie ma z
nim żadnych punktów wspólnych.
Twierdzenie 6
Prosta jest rozłączna z okręgiem wtedy i tylko
wtedy, gdy odległość środka okręgu od tej prostej jest większa od promienia
okręgu.
![](data:image/png;base64,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)
Wzajemne położenie dwóch okręgów:
Twierdzenie 7
Okręgi
i
są
rozłączne zewnętrznie wtedy i tylko wtedy, gdy
.
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)
Okręgi nie mają wtedy żadnych punktów wspólnych.
Twierdzenie 8
Okręgi
i
są styczne zewnętrznie wtedy i tylko wtedy,
gdy
.
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Okręgi mają wtedy jeden punkt wspólny.
Twierdzenie 9
Okręgi
i
przecinają się wtedy i tylko wtedy, gdy
.
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Okręgi mają wtedy dokładnie dwa punkty wspólne.
Twierdzenie 10
Okręgi
i
są styczne wewnętrznie wtedy i tylko wtedy,
gdy
.
![](data:image/png;base64,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)
Okręgi mają wtedy jeden punkt wspólny i jeden z okręgów
znajduje się w drugim.
Twierdzenie 11
Okręgi
i
są rozłączne wewnętrznie wtedy i tylko wtedy,
gdy ![](data:image/png;base64,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)
![](data:image/png;base64,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Okręgi nie mają wtedy żadnych punktów wspólnych i jeden okrąg znajduje się w drugim.
Okręgi współśrodkowe to okręgi mające wspólny środek. Jeśli te okręgi mają jednakowe promienie to się pokrywają.
Przykład 1
Dwa okręgi
i
są styczne zewnętrznie do siebie i
wewnętrznie do okręgu
Oblicz promień największego okręgu, wiedząc,
że obwód trójkąta
jest równy 12 cm.
![](data:image/png;base64,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)
Wiemy, że
. Oprócz tego:
, bo okręgi są styczne
zewnętrznie
, bo okręgi są styczne
wewnętrznie
, bo okręgi są styczne wewnętrznie
Zatem:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Odp. Promień największego okręgu ma długość 6 cm.
Zadania do zrobienia
1. W kole poprowadzono średnicę
i cięciwę
. Wiedząc, że
i
, oblicz odległość
cięciwy
od środka koła.
Odp. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACoAAAARCAMAAACcoM4HAAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OgBmOjpmOmaQOma2OpDbZgAAZgA6ZjoAZmZmZma2ZpDbZrbbZrb/kDoAkNv/tmYAtmY6tmZmtmaQttvbttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2/9vb//+2///bq4D7FQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAz0lEQVQoU81P2RKCMAxMEcULPIACXgUEJf//g6YJFF8YfLQz22aazWYX4O/O9Tlp6V0o5WfcNorOcpKKOgbU0jfTNLep58xTsdrexcCXantUKkTtZVXgP0pC73An/iqy2ttugwRMernpU9JFFqlYeEVODvOFnUe94k4XhYKDBAdovKGCdm1LuendPBiN02pYSppCpfbg3iYwdkkXJbyO3KCmMud5O1CfAfKQgTosSaGmqBTLUouAS5ZTas+fDCi8eFg7rhx/JqvCBfuBPEf5ABayEQ4K0yOpAAAAAElFTkSuQmCC)
2. Dwa okręgi są styczne zewnętrznie.
Odległość między ich środkami wynosi
. Wyznacz promienie tych okręgów, wiedząc, że:
a) jeden z nich jest o
dłuższy od drugiego
b) jeden z nich jest trzy razy dłuższy od
drugiego
Odp. a)
![](data:image/png;base64,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)
b)
![](data:image/png;base64,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)
3. Promienie dwóch okręgów
i
są równe odpowiednio:
i
. Gdyby te okręgi były
styczne zewnętrznie, to odległość między ich środkami wynosiłaby
; a gdyby te okręgi
były styczne wewnętrznie, to odległość między ich środkami byłaby równa
. Oblicz promienie tych
okręgów.
Odp. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEMAAAARCAMAAACxUuG9AAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZma2ZpBmZpC2ZpDbZrbbZrb/kDoAkGY6kNv/tmYAtmY6tmZmttvbttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bZvwztgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA/ElEQVQ4T+1S2RKCMAxs8aoX9T4q4IEI2P7/95mkdAa1wIOvZiZ0mO4m200Y+8enAyYRo3ZXzInzwbUFU4p53lFCjXITD5tRWh67RlOOI8bo0xBprX655Dw0KogyMbjdIS2H6K6XB2PUdMX5zGLFgaXH80WtD1piVgq1nOXsIejPh9ES3Mg43htlrdUytLlw6gvB+1uq4cVQM6NCJxnPyY2yeHOx6OHLnCtvGGzoagAPAx2irE+8EoDcb0wMg88CVI09HhvGYqgKCeLukfPkebKifBjQkMD+2IenYG5OAyBqEuzdNOJgV22HB9O5GpX2LlzbfdK8WL+U/eK+ACNRHcKoC9ANAAAAAElFTkSuQmCC)
4. Dane są takie dwa okręgi
że:
a)
,
,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABsAAAARCAMAAAAbkScBAAAAAXNSR0IArs4c6QAAAFFQTFRFAAAAAAAAAAA6AABmADqQAGa2OgA6OgBmOma2OpDbZgAAZgA6ZrbbZrb/kDoAkNv/tmYAtmY6tv/btv//25A627Zm2////7Zm/9uQ//+2///bsB2u0QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAXUlEQVQoU2NgGHggycXEj9MV4ixiOOVEWHE7XoCXQZQRKC/FxwgBCDskufkZRDixa5VgFxLkwWGqCCMjszBIDouZApwSbDg8IcnFK8XHKcqLzVQJdmEGEUYO2sQBAGJSAxm+kGaFAAAAAElFTkSuQmCC)
b)
,
, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD4AAAARCAMAAACy5e+VAAAAAXNSR0IArs4c6QAAAGlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjqQOma2OpC2OpDbZgAAZjqQZrbbZrb/kDoAkDo6kDpmkNv/tmYAtmY6tv//25A627Zm27aQ2////7Zm/9uQ/9vb//+2///b/seo2gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA50lEQVQ4T9WS2xKCMAxEU7yhogioLSit5P8/0iRYrSMUn5wxD6VTctjdFIC/Lrdte/8uLfmhFVVC2+eLoXg3k6rFmSmPayV4l5eAZlZHcaz2gNWShD1ud3TEn9vUAHYCF0MNNXkcT7UW3K4Arilto+YFD9SbEnqcs7+nIps8Dx4Jp/XFKb1Id2hBky61UnQdphr5MfBYSFYZncybcbcmBY4+YR6NeO27ZJEslpdwKOLo0/yDFpIdAxghycKFJxdXtxkDZJRwNCprwaqkAJeT0DxINZy84zYVjTh5ca/sQxo/wUeu9bvjO11uEoCt8b14AAAAAElFTkSuQmCC)
Odp. a)
; stąd
+
); jeśli
, 1), to okręgi są
rozłączne wewnętrzenie; jeśli
, to okręgi są styczne
wewnętrznie; jeśli
to okręgi się przecinają; jeśli
, to okręgi są styczne
zewnętrznie; jeśli
(5, +
), to okręgi są
rozłączne zewnętrznie.
b)
0 i
0, stąd
; jeśli
, to okręgi są styczne
wewnętrznie; jeśli
∈ (-1, 1) ∪
(3, 5), to okręgi są rozłączne wewnętrznie; jeśli
, to okręgi się
przecinają.