Circle Theorem VDO, Notes, Worksheet

 

Circle Theorems

A Comprehensive Notes for Students

Circle theorems are fundamental concepts in geometry that describe the relationships between angles, lines, and segments within a circle. Mastering these theorems is crucial for excelling in geometry and related fields.

Key Theorems and Concepts:

1. Angle at the Centre Theorem

Theorem: The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the remaining part of the circle.

This means if you have an angle formed at the center of the circle by two radii, and another angle formed by the same arc at a point on the circumference, the angle at the center will always be double the angle at the circumference.

2. Angles in the Same Segment Theorem

Theorem: Angles subtended by the same arc in the same segment of a circle are equal.

Imagine drawing several angles from the same two points on the circumference to other points on the circumference, all within the same segment. All these angles will have the same measure.

3. Angle in a Semicircle Theorem

Theorem: The angle subtended by a diameter at any point on the circumference is a right angle (90 degrees).

This is a special case of the Angle at the Centre Theorem, where the angle at the center is 180 degrees (a straight line), so the angle at the circumference is half of that, which is 90 degrees.

4. Cyclic Quadrilateral Theorem

Theorem: The opposite angles of a cyclic quadrilateral (a quadrilateral whose vertices all lie on the circumference of a circle) add up to 180 degrees.

eorem is very useful for finding unknown angles in figures involving circles and quadrilaterals.

5. Tangent-Radius Theorem

Theorem: The tangent to a circle at any point is perpendicular to the radius through the point of contact.

This creates a right angle where the tangent touches the circle, which is often helpful in solving problems involving tangents.

6. Alternate Segment Theorem

Theorem: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

 theorem can be a bit tricky but is extremely powerful for finding angles in complex circle diagrams involving tangents.

7. Intersecting Chords Theorem

Theorem: If two chords intersect inside a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord.

This theorem helps relate the lengths of segments created by intersecting chords.

8. Intersecting Secants Theorem

Theorem: If two secant segments are drawn to a circle from an exterior point, then the product of the length of one secant segment and its external segment is equal to the product of the length of the other secant segment and its external segment.

This theorem deals with lengths when two lines intersect outside the circle.

9. Tangent-Secant Theorem

Theorem: If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment is equal to the product of the length of the secant segment and its external segment.

This theorem combines elements of tangents and secants, providing a powerful tool for solving problems involving both.

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