### Circle Theorems  More

Year 10/11 maths revision: terms for a circle: centre, radius, chord, diameter, circumference, tangent, arc, sector, segment and subtended.

For a high grade you need to know the proofs for the circle theorems.

###### Circle basics  Menu

The 'radius' of a circle is the distance from the centre to its edge.

The 'diameter' of a circle is the distance between one edge of a circle, through its centre, to the edge on the other side.
The diameter of a circle is twice the value of its radius.

The 'circumference' is the distance around the edge of a circle. The circumference of a circle can be calculated by the formula: 2πr or πD

The 'area' of a circle can be calculated by the formula: πr²

The 'tangent' to a circle is a line which touches a circle at one point; without cutting across the circle.

An 'arc' is part of the circumference of a circle.

A 'chord' a line which goes from one point to another on the circle's circumference.

A 'sector' is a pie-slice portion of a circle, e.g. a semicircle because it is a half of the whole circle.

A 'segment' is the portion of a circle between a chord of a circle and its associated arc.

The angle x is an angle 'subtended' by the arc.

Q: A circle of diameter 10cm has an arc length π cm.
How many arcs are in a circumference? ANS

###### Angle at centre is twice that at circumference  Menu

Prove that the angle subtended by an arc at the centre (2b) is twice the angle subtended at the circumference (b).

Hint: Draw a line from the centre o, to the point where the two lines cross the circumference at the top of the diagram

This makes two isosceles triangles with sides equal to the radius.

ANS

###### Angle in semi-circle is 90°  Menu

Prove that the angle c, which is subtended at the circumference by a semicircle is 90° or a right angle.

Hint: use the proof above for 'Angle at centre is twice that at circumference', with the angle at the centre being 180°

ANS
###### Angles within the same segment are equal Menu

The angles at the circumference subtended by the same arc are equal or
angles in the same segment are equal.

Prove that angle d and angle e are equal

Hint: draw a line from P to centre O and then to Q

ANS

###### Opposite angles in cyclic quadrilateral add up to 180° Menu

A cyclic quadrilateral has four sides with each corner touching the inside circumference of a circle.

Prove that the opposite angles in a cyclic quadrilateral add up to 180°.

i.e. angle m + n = 180°

Hint: draw a line from to the centre O from the left and right corner of the shape

ANS

###### The angle between a tangent and a radius in a circle is 90°. Menu

The line that just touches a circle is called a tangent.

Prove that the angle between a tangent and a radius in a circle is 90°.

ANS

###### The lengths of the two tangents from a point to a circle are equal. Menu

Prove that the lengths of the two tangents from a point (B) to a circle are equal

You know that ∠BAO, ∠BCO are both right angles from the previous proof.

Hint: Draw the line OB.

ANS

###### Perpendicular from the centre to chord bisects the chord. Menu

Prove that the perpendicular from the centre of a circle C to a chord AB bisects the chord.

A perpendicular to the chord is 90°

Hint: Draw a line from C to A and C to B to make two triangles

ANS

###### Alternate segment theorem 1 Menu

Prove that the angle between the tangent and the chord (p) is equal to the angle in the alternate segment (q).

Hint: the line AC goes through the centre O so this is a straight line. Angle in a semi-circle?

ANS

###### Alternate segment theorem 2 Menu

Another example of the alternate segment theorem

Prove that the angle between the tangent and the chord (s) is equal to the angle in the alternate segment (q).

Hint: draw two lines from the A and C to O to make a triangle

ANS

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