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.




Basics and 8 circle theorems
Circle basics  Menu

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

diameter

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

tangent chord sector segment

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
Angle at centre is twice that at circumference

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
angle in semi-circle is 90 degrees

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
angle in same segment are equal

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
Opposite angles in cyclic quadrilateral add up to 180

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
tangent to circle is 90 degrees

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
tangent to circle is 90 degrees

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
chord bisect

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
alternate segment

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
alternate segment

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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