Euclidean Geometry (Grade 11) — Mathematics Grades 8–11

The circle theorems: chords and the centre, angles in the circle, cyclic quadrilaterals and tangents.

Gr 11 Term 3 - Euclidean GeometryTerm 3 - 3 weeks (CAPS)

What gets asked

Name the theorem as the reason every time an angle or length moves.
The angle at the centre is DOUBLE the angle at the circle on the same arc.
Opposite angles of a cyclic quadrilateral add to 180 degrees.
A tangent meets the radius at 90 degrees, and two tangents from a point are equal.

Equating angles subtended by the same chord but on opposite sides. Same segment means the same side of the chord; opposite sides are supplementary via the cyclic quad.
Halving the wrong angle in the centre theorem. The CENTRE angle is the big one: angle at centre = twice angle at circumference.
Using the whole chord in the right triangle made by the perpendicular from the centre. The perpendicular BISECTS the chord - the triangle uses half of it.

O

is the centre.

A

B

and

C

lie on the circle with

\hat{A O B} = 9 6^{\circ}

C

on the major arc. Find

\hat{A C B}

The angle at the centre is double the angle at the circle on the same arc $A B$ .
$\hat{A C B} = 9 6^{\circ} \div 2 = 4 8^{\circ}$ .