Circle theorems - Exam questions

Circle theorems - Exam questions

Circle Theorems Revision Content

Theorem 1: Angles Subtended by the Same Arc

• The angles subtended by the same arc or chord of a circle at the circumference are equal.
• This can be applied to triangles formed in a circle where the base is the chord of the circle. If a triangle has vertices on the circumference of a circle and the base of the triangle is a chord of the circle, then the angles of the triangle subtended by that chord are equal.

Theorem 2: The Angle at the Centre verses the Angle at the Circumference

• The angle subtended by an arc at the centre of the circle is twice the angle subtended by the same arc at any point on the alternate segment of the circle.
• When a triangle is inscribed in a circle with one vertex at the center of the circle, the angle at the center of the circle is twice the angle at the circumference subtended by the same arc.

Theorem 3: Opposite Angles of a Cyclic Quadrilateral

• Opposite angles of a cyclic quadrilateral add to 180 degrees.
• A quadrilateral is cyclic if a circle can be drawn around it, touching all four vertices. In such a quadrilateral, the opposite angles are supplementary.

Theorem 4: Perpendicular from the Centre to a Chord

• The line drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord.
• The triangle formed by a chord and the perpendicular line from the centre of the circle to the chord is a right-angled triangle. The right-angle is at the intersection of the chord and the perpendicular line.

Theorem 5: Tangent to a Circle

• The tangent to a circle forms a right angle with the radius of the circle at the point of contact.
• If a right-angled triangle is drawn from the centre of a circle to a tangent, then the hypotenuse will be the radius, and the side adjacent to the right angle will be the segment of the tangent line extending from the circle to the triangle.

Theorem 6: Alternate Segment Theorem

• The angle between a tangent and a chord of a circle equals the angle in the opposite segment.
• This theorem can be applied by constructing two triangles inside the circle, both sharing a common side that is the chord and one having the other side as the tangent. The angles subtended by the chord in the segment not containing the tangent are equal to the angle between the tangent and the chord.

Remember, understanding and practising these theorems are important for solving geometry problems related to circles.