Geometry Study Guide

A locus is a set of all points that satisfy a given condition or conditions. In simple terms, it's the path traced by a point moving according to a rule. For GCSE Mathematics, you'll typically encounter three main types of loci: equidistant from a fixed point, equidistant from two fixed points, and equidistant from two intersecting lines.

1. Locus of points equidistant from a fixed point: This describes all points that are the same distance away from a central point. The shape formed by such points is a circle, where the fixed point is the centre and the fixed distance is the radius.

1. Locus of points equidistant from two fixed points: This describes all points that are the same distance from two different fixed points. The resulting shape is the perpendicular bisector of the line segment connecting the two fixed points. The perpendicular bisector cuts the line segment exactly in half and at a 90-degree angle.

1. Locus of points equidistant from two intersecting lines: This describes all points that are the same distance from two lines that cross each other. The shape formed is the angle bisector of the angle between the two lines. The angle bisector divides the angle into two equal parts.

To construct these loci, you will use a compass and a straightedge (ruler). Accuracy is key. For a circle, place the compass point on the fixed point and open it to the desired radius. For a perpendicular bisector, set the compass to a radius more than half the distance between the two points, draw arcs from both points intersecting above and below the line, then draw a straight line through these intersections. For an angle bisector, draw an arc from the vertex that intersects both lines, then from these intersection points, draw two more arcs that intersect inside the angle, finally draw a line from the vertex through this intersection.

Angles around a point add up to 360 degrees. Imagine standing in the centre of a circle and turning completely around. You would have made a full rotation of 360 degrees. Similarly, if you have several angles that meet at a single point, and they collectively form a full turn, their sum must always be 360 degrees. This fundamental geometric principle is crucial for solving many problems involving angles. For example, if you know the values of some angles around a point, you can find the missing angle by subtracting the known angles from 360 degrees. This concept can be applied whether there are two, three, or more angles at that point. It's important to recognise these situations in diagrams, even if the angles aren't explicitly drawn as a full circle, as long as they all share a common vertex and cover all directions around it.