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•Victory •
@victory_ebof
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Okrąg i cięciwy - kluczowe pojęcia i obliczenia w geometrii
26.04.2022
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Page 2: Problem Solving with Circle Geometry
This page focuses on applying circle geometry concepts to solve practical problems. It presents two main problems involving chord lengths and distances from the center of a circle.
Problem 4.43: Given a circle with a radius of 25 cm and a chord AB with length 48 cm, calculate the distance of the chord from the center of the circle.
Example: The solution uses the Pythagorean theorem to find the unknown distance: x² + 24² = 25² x² = 625 - 576 = 49 x = 7 cm
Problem 4.44: For a circle with a radius of 17 cm, calculate the length of a chord that is 8 cm away from the center of the circle.
Example: Again, the Pythagorean theorem is applied: x² + 8² = 17² x² = 289 - 64 = 225 x = 15 cm Chord length = 2x = 30 cm
Highlight: These problems demonstrate the practical application of the Pythagorean theorem in solving circle geometry questions.
Page 3: Additional Circle Properties and Practice Problems
This page introduces additional properties of circles and provides practice problems to reinforce understanding.
Vocabulary: Chord - A line segment whose endpoints lie on a circle.
Vocabulary: Arc - A portion of the circumference of a circle.
The page presents a problem (4.41) asking students to sketch a circle and mark points on it:
a) Determine the number of chords and arcs formed by three points on a circle. b) Determine the number of chords and arcs formed by four points on a circle.
Highlight: This problem helps students visualize how points on a circle create various geometric elements.
The page also mentions that if tangent lines are drawn from an external point to a circle, the lengths of these tangent lines are equal. This property is illustrated with a diagram.
Example: A diagram shows a circle with an external point P and two tangent lines PA and PB, emphasizing that |PA| = |PB|.
These concepts and problems help students develop a deeper understanding of circle geometry and the mutual positions of lines and circles.
Page 4: Practice Problems and Applications
This final page provides additional practice problems to reinforce the concepts learned about circles and their properties.
Problem 4.43 (Repeated from Page 2): This problem involves calculating the distance of a chord from the center of a circle, given the radius and chord length.
Highlight: This repetition emphasizes the importance of applying the Pythagorean theorem in circle geometry problems.
Problem 4.44 (Repeated from Page 2): This problem asks students to calculate the length of a chord given its distance from the center and the circle's radius.
Example: These problems demonstrate practical applications of circle geometry and the mutual position of a line and circle.
The repetition of these problems underscores their significance and provides students with additional opportunities to practice these important concepts.
Vocabulary: Mutual position - The relative arrangement of geometric objects, in this case, a line and a circle.
By working through these problems, students can improve their understanding of circle geometry, the Pythagorean theorem, and their applications in real-world scenarios.
Page 1: Circle Basics and Line-Circle Positions
This page introduces fundamental concepts related to circles and the mutual positions of lines and circles.
Definition: A circle is a set of points equidistant from a central point.
Key circle elements are defined: • S - Center of the circle • r - Radius of the circle • AB - Diameter of the circle (AB = 2r)
The page then explains the three possible mutual positions of a line and circle:
Highlight: The distance between the center of the circle and a line determines its position relative to the circle.
Example: A diagram illustrates each of these positions, showing how the distance from the center (|SP|) compares to the radius (r) in each case.
The page concludes with a note about tangent lines drawn from an external point, stating that the lengths of these tangent lines are equal.
19
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1/2
koło i okrąg
koło i okrąg klasa 1
76
1864
4
okrąg w układzie współrzędnych
równanie okręgu, przykłady zadań
235
4824
1/2
Koła i okręgi
Koła i okręgi matma
955
19352
1/2
geometria płaska - okręgi i koła
geometria płaska - okręgi i koła
90
2708
1/2
Kąty w okręgu-zadania
Kąty wpisane i środkowe w okręgu w zadaniach . Treści zadań -źródło: matemaks.pl
81
2042
4/5
Geometria analityczna - zadania maturalne cz 1
10 zadań maturalnych z różnych lat
Średnia ocena aplikacji
Uczniowie korzystają z Knowunity
W rankingach aplikacji edukacyjnych w 12 krajach
Uczniowie, którzy przesłali notatki
Użytkownik iOS
Filip, użytkownik iOS
Zuzia, użytkownik iOS
•Victory •
@victory_ebof
·
13 Obserwujących
Obserwuj
Ekspert przedmiotu
Okrąg i cięciwy - kluczowe pojęcia i obliczenia w geometrii
Dostęp do wszystkich materiałów
Popraw swoje oceny
Dołącz do milionów studentów
Rejestrując się akceptujesz Warunki korzystania z usługi i Politykę prywatności.
Page 2: Problem Solving with Circle Geometry
This page focuses on applying circle geometry concepts to solve practical problems. It presents two main problems involving chord lengths and distances from the center of a circle.
Problem 4.43: Given a circle with a radius of 25 cm and a chord AB with length 48 cm, calculate the distance of the chord from the center of the circle.
Example: The solution uses the Pythagorean theorem to find the unknown distance: x² + 24² = 25² x² = 625 - 576 = 49 x = 7 cm
Problem 4.44: For a circle with a radius of 17 cm, calculate the length of a chord that is 8 cm away from the center of the circle.
Example: Again, the Pythagorean theorem is applied: x² + 8² = 17² x² = 289 - 64 = 225 x = 15 cm Chord length = 2x = 30 cm
Highlight: These problems demonstrate the practical application of the Pythagorean theorem in solving circle geometry questions.
Dostęp do wszystkich materiałów
Popraw swoje oceny
Dołącz do milionów studentów
Rejestrując się akceptujesz Warunki korzystania z usługi i Politykę prywatności.
Page 3: Additional Circle Properties and Practice Problems
This page introduces additional properties of circles and provides practice problems to reinforce understanding.
Vocabulary: Chord - A line segment whose endpoints lie on a circle.
Vocabulary: Arc - A portion of the circumference of a circle.
The page presents a problem (4.41) asking students to sketch a circle and mark points on it:
a) Determine the number of chords and arcs formed by three points on a circle. b) Determine the number of chords and arcs formed by four points on a circle.
Highlight: This problem helps students visualize how points on a circle create various geometric elements.
The page also mentions that if tangent lines are drawn from an external point to a circle, the lengths of these tangent lines are equal. This property is illustrated with a diagram.
Example: A diagram shows a circle with an external point P and two tangent lines PA and PB, emphasizing that |PA| = |PB|.
These concepts and problems help students develop a deeper understanding of circle geometry and the mutual positions of lines and circles.
Dostęp do wszystkich materiałów
Popraw swoje oceny
Dołącz do milionów studentów
Rejestrując się akceptujesz Warunki korzystania z usługi i Politykę prywatności.
Page 4: Practice Problems and Applications
This final page provides additional practice problems to reinforce the concepts learned about circles and their properties.
Problem 4.43 (Repeated from Page 2): This problem involves calculating the distance of a chord from the center of a circle, given the radius and chord length.
Highlight: This repetition emphasizes the importance of applying the Pythagorean theorem in circle geometry problems.
Problem 4.44 (Repeated from Page 2): This problem asks students to calculate the length of a chord given its distance from the center and the circle's radius.
Example: These problems demonstrate practical applications of circle geometry and the mutual position of a line and circle.
The repetition of these problems underscores their significance and provides students with additional opportunities to practice these important concepts.
Vocabulary: Mutual position - The relative arrangement of geometric objects, in this case, a line and a circle.
By working through these problems, students can improve their understanding of circle geometry, the Pythagorean theorem, and their applications in real-world scenarios.
Dostęp do wszystkich materiałów
Popraw swoje oceny
Dołącz do milionów studentów
Rejestrując się akceptujesz Warunki korzystania z usługi i Politykę prywatności.
Page 1: Circle Basics and Line-Circle Positions
This page introduces fundamental concepts related to circles and the mutual positions of lines and circles.
Definition: A circle is a set of points equidistant from a central point.
Key circle elements are defined: • S - Center of the circle • r - Radius of the circle • AB - Diameter of the circle (AB = 2r)
The page then explains the three possible mutual positions of a line and circle:
Highlight: The distance between the center of the circle and a line determines its position relative to the circle.
Example: A diagram illustrates each of these positions, showing how the distance from the center (|SP|) compares to the radius (r) in each case.
The page concludes with a note about tangent lines drawn from an external point, stating that the lengths of these tangent lines are equal.
Matematyka - koło i okrąg
koło i okrąg klasa 1
19
1275
0
Matematyka - okrąg w układzie współrzędnych
równanie okręgu, przykłady zadań
76
1864
2
Matematyka - Koła i okręgi
Koła i okręgi matma
235
4824
4
Matematyka - geometria płaska - okręgi i koła
geometria płaska - okręgi i koła
955
19352
25
Matematyka - Kąty w okręgu-zadania
Kąty wpisane i środkowe w okręgu w zadaniach . Treści zadań -źródło: matemaks.pl
90
2708
4
Matematyka - Geometria analityczna - zadania maturalne cz 1
10 zadań maturalnych z różnych lat
81
2042
7
Średnia ocena aplikacji
Uczniowie korzystają z Knowunity
W rankingach aplikacji edukacyjnych w 12 krajach
Uczniowie, którzy przesłali notatki
Użytkownik iOS
Filip, użytkownik iOS
Zuzia, użytkownik iOS
