A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. Let's explore the key parts of a circle. The center is the fixed point, shown here as point O. The radius is the distance from the center to any point on the circle. The diameter is the longest chord that passes through the center. A chord is any line segment joining two points on the circle. An arc is a part of the circle's circumference. Finally, a tangent is a line that touches the circle at exactly one point.
Now let's explore the important properties of chords in a circle. The first theorem states that a perpendicular from the center to a chord bisects the chord. This means it cuts the chord into two equal parts. The second theorem tells us that equal chords are equidistant from the center. The third theorem is the converse: chords that are equidistant from the center are equal in length. These properties are fundamental for solving many circle problems.
Let's explore angles in circles. A central angle is formed by two radii meeting at the center of the circle. An inscribed angle is formed by two chords that meet at a point on the circle. The inscribed angle theorem is one of the most important theorems in circle geometry. It states that an inscribed angle is always half the central angle when both angles subtend the same arc. This relationship is fundamental for solving many circle problems.
Now let's examine the important properties of tangents. The tangent-radius theorem states that a tangent to a circle is always perpendicular to the radius at the point of contact. This creates a right angle between the tangent line and the radius. The tangent segment theorem tells us that if we draw tangent segments from an external point to a circle, these segments are equal in length. This property is very useful for solving problems involving tangents from external points.
To summarize what we've learned about circles: A circle is defined as the set of all points equidistant from a center. A perpendicular from the center to any chord bisects that chord. Inscribed angles are always half the measure of their corresponding central angles. Tangent lines are perpendicular to radii at their points of contact. These fundamental properties form the foundation for solving complex geometric problems involving circles.