Example 1: Basic Definitions (Radius, Diameter, Chord, Tangent)
Define key terms related to a circle.
Circle: A set of all points in a plane that are at a fixed distance from a fixed point in the plane. The fixed point is the center, and the fixed distance is the radius.
Radius: A line segment connecting the center of the circle to any point on the circle.
Diameter: A chord passing through the center of the circle. It is the longest chord and is equal to twice the radius (Diameter = 2 * Radius).
Chord: A line segment joining any two points on the circle.
Tangent: A line that intersects the circle at exactly one point. This point is called the point of contact.
Secant: A line that intersects the circle at two distinct points.
Example 2: Tangent Perpendicular to Radius Theorem
State and explain the theorem about the tangent at any point of a circle.
Theorem: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Explanation: If a line AB is tangent to a circle at point P, and O is the center of the circle, then the radius OP is perpendicular to the tangent AB at P. This means Angle OPA = 90 degrees.
Converse: A line drawn through the end point of a radius and perpendicular to it is a tangent to the circle.
Example 3: Tangents from an External Point Theorem
State and explain the theorem about the lengths of tangents drawn from an external point.
Theorem: The lengths of tangents drawn from an external point to a circle are equal.
Explanation: If P is an external point and PA and PB are two tangents drawn from P to a circle with center O, touching the circle at points A and B respectively, then PA = PB.
Proof Idea: You can prove this by considering triangles OAP and OBP. Both are right-angled triangles (by the Tangent-Radius Theorem). OA = OB (radii), OP is common. By RHS congruence, Triangle OAP is congruent to Triangle OBP. Therefore, PA = PB (by CPCT - Corresponding Parts of Congruent Triangles).
Example 4: Angle Subtended by an Arc at the Center and Circumference
State and explain the theorem relating the angle subtended by an arc at the center and at a point on the remaining part of the circle.
Theorem: The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
Explanation: If AB is an arc of a circle with center O, and C is any point on the remaining part of the circle, then Angle AOB = 2 * Angle ACB.
Special Case: The angle in a semicircle is a right angle (90 degrees). This is because the diameter subtends a straight angle (180 degrees) at the center, and half of 180 is 90.
Example 5: Angles in the Same Segment
State and explain the theorem about angles in the same segment of a circle.
Theorem: Angles in the same segment of a circle are equal.
Explanation: If an arc AB subtends angles at two points C and D on the remaining part of the circle, then Angle ACB = Angle ADB. These angles are in the same segment (the major segment formed by chord AB).
Practice Mode
Enter basic circle properties to calculate related values.
Note: This solver handles basic radius/diameter calculations and simple tangent length problems.
Related Concepts
Explore these related mathematical concepts to deepen your understanding of Circles.
Center
The fixed point from which all points on the circle are equidistant.
Circumference
The distance around the circle.
Area
The amount of space enclosed by the circle.
Arc
A continuous part of the circumference of a circle.
Sector
The area enclosed by two radii and an arc of a circle.
Segment
The area enclosed by a chord and an arc of a circle.
Cyclic Quadrilateral
A quadrilateral whose all four vertices lie on the circumference of a circle.
Concentric Circles
Circles that have the same center but different radii.