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:

1. Tangent line - The line touches the circle at exactly one point
2. Secant line - The line intersects the circle at two points
3. External line - The line has no points in common with the 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.