Circle Theorems (10A)

Prove and apply angle and chord properties of circles (VCMMG366)

LO: To prove and apply angle and chord properties of circles

Know:

  • The properties of a circle
  • Vocabulary of a circle, such as arc, centre, circumference, radius, diameter, chord, sector and segment.

Understand:

  • That circles have certain angular properties.

Do:

  • I can prove and apply angle and chord properities in circles.

Angle Properties in Circles

Inscribed angle is an angle created from any 2 points on the circumference of a circle meeting on a 3rd point on the circumference.  The central angle is an angle created from any 2 points on the circumference of a circle meeting at the centre of the circle.

The inscribed angle is always half the size of the central angle.

BBC Bitesize

An angle inscribed in a semicircle is always a right angle.

BBC Bitesize

Angles in the same segment (same 2 starting points on the circumference) are always equal.

BBC Bitesize

A cyclic quadrilateral is a quadrilateral (4-sided shape) whose vertices all touch the circumference of a circle.

Opposite angles in a cyclic quadrilateral add up to 180 degrees or is supplementary.

BBC Bitesize

The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.

BBC Bitesize

Chord Properties in Circles

A chord is a straight line from one point on the circumference of a circle to another.

If a chord is bisected (split evenly) by the radius of a circle, then both sides of the chord are equal in length.

It also forms a 90 degree angle between the chord and the radius.

BBC Bitesize

Chords that are equal in length are equidistant (same distance) from the circle centre.

When two chords intersect inside a circle, this divides each chord into two line segments in which the product of the lengths of the line segments for both chords is the same.

Tangent and Secants

A secant is a line that cuts a circle twice.

A tangent is a line that touches the circumference of a circle at one point only.

BBC Bitesize

A tangent that touches a radius of the circle always creates a 90 degree angle (right angle).

If two tangents intersect outside of a circle, the distances along the tangent from the intersection to the circumference of the circle are equal.

If two secants intersect outside a circle, the product of the entire secant length by the external secant length will be the same.

If a tangent and a secant intersect outside a circle, the product of the entire secant length by the external secant length will be equal to the square of the tangent length.

Circle Properties Videos

Inscribed Angles Videos

Circle Theorem

Chord Product Theorem

Secant Product Theorem

Tangent Product Theorem

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