Circle theorems and constructionsIB MYP Maths Extended: Revision notes

A chord is a straight line joining two points on the circle. A diameter is a chord through the centre. An arc is part of the circumference. A tangent is a straight line that touches the circle at exactly one point. Every radius is the same length, so a triangle made from two radii and a chord is isosceles. This is used in many circle theorem questions.

The angle at the centre of a circle is twice the angle at the circumference, when both are on the same arc. If angle $AOB=116^\circ$ at the centre, the angle at the circumference is $58^\circ$. A special case is the angle in a semicircle: the angle at the circumference standing on a diameter is $90^\circ$. If $PQ$ is a diameter and $R$ is on the circle, then angle $PRQ=90^\circ$.

Angles at the circumference on the same arc, on the same side of the chord, are equal. If $C$ and $E$ are on the major arc $AB$, then angle $ACB$ equals angle $AEB$. These are called angles in the same segment. A segment is the part of a circle cut off by a chord.

A cyclic quadrilateral has all four vertices on a circle. Its opposite angles add up to $180^\circ$. If $A$, $C$, $B$, $D$ lie on a circle in that order, angle $ACB+$ angle $ADB=180^\circ$ when $C$ and $D$ are on opposite arcs of the chord $AB$. For example $ACB=58^\circ$ gives $ADB=122^\circ$.

A tangent is perpendicular to the radius at the point where it touches. Two tangents from the same outside point are equal in length. If $TA$ and $TB$ are tangents, then $TA=TB$, triangle $TAB$ is isosceles, and the angles $OAT$ and $OBT$ are both $90^\circ$. Angles in the quadrilateral $OATB$ add up to $360^\circ$.

In circle theorem questions every step needs a reason. Use short, exact wording: 'angle at the centre is twice the angle at the circumference', 'angle in a semicircle is $90^\circ$', 'angles in the same segment are equal', 'opposite angles of a cyclic quadrilateral add up to $180^\circ$', 'tangent perpendicular to radius', 'tangents from an external point are equal', 'radii are equal so the triangle is isosceles'. Also name the angle with three letters so the marker knows which angle you mean.

To construct the perpendicular bisector of a line $AB$: open the compasses to more than half of $AB$, draw arcs from $A$ and $B$ so they cross above and below, then join the two crossing points. Every point on this line is the same distance from $A$ and $B$. To construct the angle bisector of an angle at $V$: draw an arc from $V$ that cuts both arms, draw equal arcs from those two points so they cross, then draw a line from $V$ through the crossing point. Do not rub out the construction arcs. The centre of a circle through three points is where the perpendicular bisectors of two chords meet.