Geometry and Trigonometry 2

Section 1Circle theorem recap

Circle geometry questions are really angle-chasing questions. Use the given markings first: equal radii, tangents, chords, diameters, and points on the circumference. Watch out: a theorem only works when its conditions are actually shown or stated.

Section 2Cyclic quadrilaterals

A cyclic quadrilateral has all four vertices on a circle. Its opposite angles add to 180 degrees. Here's the trick: if one angle is 74 degrees, the opposite angle is 106 degrees, not another 74 degrees.

Section 3Angles in the same segment

Angles standing on the same chord and in the same segment are equal. In plain words, if two angles touch the same chord from the same side of the circle, they match. This is a common way to prove two angles are equal without measuring them.

Section 4Tangent-chord theorem

The angle between a tangent and a chord equals the angle in the alternate segment. Easy to mix up - the matching angle is inside the circle on the opposite arc, not the angle right beside the tangent unless the diagram says so.

Section 5Loci

A locus is the set of all points that satisfy a rule. Points a fixed distance from one point form a circle. Points equidistant from two fixed points lie on the perpendicular bisector of the line joining them.

Section 6Constructions

Constructions use a ruler and compass to draw accurate geometric objects, not just neat sketches. Common constructions include perpendicular bisectors, angle bisectors, perpendiculars, parallel lines, and triangles from given sides or angles. Keep compass arcs visible in exam working because they show your method.

Section 7Trigonometric applications

Applied trig questions may combine bearings, elevation, scale diagrams, Pythagoras, sine, cosine, tangent, and sometimes the sine rule or cosine rule for non-right triangles. Start by drawing or cleaning up the diagram. The thing to remember is: choose the rule from the sides and angles you actually know.