Topic 17 of 18Cambridge O Levels

Loci & Constructions

Constructing paths and regions that satisfy given geometric rules using a ruler and compass.

### Introduction to Loci

In geometry, a locus (plural: loci) is a set of all points that satisfy one or more given conditions. Think of it as the path traced by a point moving according to a specific rule. For the Cambridge O Level syllabus, we explore these paths in two dimensions (on a flat plane) and learn how to construct them accurately using only a ruler (or straight edge) and a pair of compasses.

### The Four Fundamental Loci

There are four key loci you must be able to construct and describe. Your construction arcs must always be left on the diagram as proof of your method.

1. The Locus of Points at a Fixed Distance from a Fixed Point

* Condition: A point moves so that it is always a constant distance, *r*, from a fixed point, P.

* Description: The locus is a circle with centre P and radius *r*.

* Construction Process:

Identify the fixed point P.

Set the radius of your compass to the required distance *r* using your ruler.

Place the sharp point of the compass on P.

Rotate the compass to draw a complete circle.

2. The Locus of Points Equidistant from Two Fixed Points

* Condition: A point moves so that it is always the same distance from two different fixed points, A and B.

* Description: The locus is the perpendicular bisector of the line segment joining points A and B.

* Construction Process:

Identify the two fixed points, A and B, and join them with a light line if necessary.

Set your compass to a radius that is clearly more than half the distance between A and B.

Place the compass point on A and draw a wide arc that goes both above and below the line segment AB.

Keeping the same compass radius, place the compass point on B and draw another arc to intersect the first one at two places.

Use your ruler to draw a straight line through these two points of intersection. This line is the perpendicular bisector.

3. The Locus of Points at a Fixed Distance from a Straight Line

* Condition: A point moves so that it is always a constant distance, *d*, from a fixed straight line, L.

* Description: The locus is a pair of parallel lines, one on each side of the line L, each at a distance *d* from L. If the locus is at the end of a line segment, it includes a semicircle.

* Construction Process:

Draw the fixed line L.

Choose two points on L, say X and Y. At each point, construct a perpendicular line.

Use your compass or ruler to mark the distance *d* along each perpendicular, on both sides of line L.

Draw a straight line through the marked points on one side of L. This line will be parallel to L.

Repeat for the other side.

4. The Locus of Points Equidistant from Two Intersecting Lines

* Condition: A point moves so that it is always the same distance from two straight lines, L1 and L2, that intersect.

* Description: The locus is the angle bisector of the angles formed by the two lines. Since intersecting lines form two pairs of vertically opposite angles, there will be two angle bisectors, which are themselves perpendicular to each other.

* Construction Process:

Let the lines L1 and L2 intersect at point O.

Place the compass point on O and draw an arc that cuts both lines, say at points P and Q.

Place the compass point on P and draw an arc inside the angle.

Keeping the same compass radius, place the compass point on Q and draw another arc to intersect the previous one.

Use your ruler to draw a straight line from the intersection point O through the point where these two arcs cross. This line is the angle bisector.

### Combining Loci and Solving Problems

Many exam questions require you to combine two or more loci to find a specific point or region that satisfies multiple conditions simultaneously. These problems often involve inequalities.

* 'Less than' or 'within' a distance: This refers to the region inside a circle.

* 'Greater than' or 'outside' a distance: This refers to the region outside a circle.

* 'Closer to point A than point B': This refers to the region on one side of the perpendicular bisector of AB. To find which side, pick a test point (like A itself) and see if it satisfies the condition. The entire region on that side of the line is the solution.

To solve these problems, you will construct all the required loci on a single diagram. The solution is the point of intersection or the shaded region that satisfies all the given rules.

Key Points to Remember

1A locus is a set of points satisfying a specific geometric rule.
2The locus of points at a fixed distance from a point is a circle.
3The locus of points equidistant from two points is their perpendicular bisector.
4The locus of points at a fixed distance from a line is a pair of parallel lines.
5The locus of points equidistant from two intersecting lines is the pair of angle bisectors.
6Problems are often solved by accurately constructing multiple loci and identifying the resulting intersection point or region.
7Construction must be done with only a ruler and compass, and all construction arcs must be shown.
8Inequalities like 'closer to' or 'less than' define regions, which should be clearly shaded.

Pakistan Example

Planning an Irrigation Canal

The Punjab government is planning a new irrigation canal. According to the plan, the canal must be built such that it is always **equidistant from the main roads connecting Faisalabad and Sargodha**. Additionally, to serve a new industrial zone, a section of the canal must be **less than 30 km from the city of Chiniot**. On a map, construct the loci to identify the valid region for the canal. This requires constructing the **angle bisector** of the two roads and a **circle** with a 30 km scaled radius around Chiniot. The valid region for the industrial zone section is the part of the angle bisector that lies inside the circle.

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