Why Loci Matter in Geometry
Ever wondered how to describe the path a moving point makes? That’s exactly what a locus does – and it shows up a lot in ICSE exams.
💡 In Simple Words: A locus is the set of all positions a point can occupy while satisfying a given condition. Think of it as a “track” that a point follows, like the line a dog runs when you pull its leash around a pole.
What Is a Locus? (Simple Definition)
A locus (pronounced "loh‑kus") is a collection of points that all meet the same rule. Imagine you have a garden hose. If you keep the water pressure the same, the water spreads out in a circle. Every point on that circle is the same distance from the faucet – that circle is a locus.
Common Locus Theorems You’ll Need
ICSE textbooks list a few handy theorems. Knowing them saves time on the board.
- Equal Distance Theorem: The set of points that are a fixed distance from a single point forms a circle.
- Perpendicular Bisector Theorem: Points equidistant from the ends of a line segment lie on the line that cuts the segment in half at a right angle.
- Angle Bisector Theorem: Points that see a given line segment under a constant angle sit on an arc of a circle.
Worked Example 1: Locus of a Point 5 cm from a Fixed Point
**Problem:** Find the locus of all points that are exactly 5 cm away from point A.
Solution: By the Equal Distance Theorem, the answer is a circle with centre A and radius 5 cm. Draw a dot for A, then use a compass set to 5 cm – the circle you get is the locus.
Worked Example 2: Locus of Points Equidistant from Two Points
**Problem:** Locate all points that are the same distance from points B and C.
Solution: That’s the Perpendicular Bisector Theorem in action. First draw segment BC. Then find its midpoint D. Finally, draw a line through D that meets BC at a right angle. Every point on that line is equally far from B and C.
Worked Example 3: Locus Making a Constant Angle with a Line
**Problem:** Find the locus of points from which a given line AB is seen under a 60° angle.
Solution: Picture the line AB as a fence. The points that look at the fence with a 60° view sit on two arcs of circles that share AB as a chord. Those arcs are called “circular arcs of constant angle.”
Quick Comparison Table
| Condition | Resulting Locus | Key Theorem |
|---|---|---|
| Fixed distance from a point | Circle | Equal Distance Theorem |
| Equal distance from two points | Perpendicular bisector (straight line) | Perpendicular Bisector Theorem |
| Constant angle subtended by a segment | Arc of a circle | Angle Bisector Theorem |
How to Tackle Locus Questions in the Exam
Here’s a short checklist you can keep in your pocket:
- Read the condition carefully – is it a distance, an angle, or a ratio?
- Match the condition to the right theorem.
- Sketch the basic figure first (point, segment, line).
- Draw the locus using a ruler, compass or by reasoning.
- Label everything – examiners love clear diagrams.
Common Mistakes to Avoid
- Mixing up “inside” and “on” the locus. The locus itself is the line or curve, not the area inside it (unless the question says “region”).
- Forgetting that a perpendicular bisector is infinite – it extends beyond the segment.
- Skipping the step of checking the condition after drawing. A quick test point can save marks.
📝 Likely Exam Questions
- Question: Find the locus of a point that moves so that it is always 3 cm from point O and also 4 cm from point P. Answer: The point must satisfy both distance conditions, so the locus is the intersection of two circles – one centred at O with radius 3 cm and the other at P with radius 4 cm. Plot both circles; the common points (usually two) are the required positions.
- Question: Determine the locus of points that are equidistant from the ends of a 10 cm line segment AB. Answer: By the Perpendicular Bisector Theorem, the locus is the line that cuts AB at its midpoint and is perpendicular to AB.
- Question: A point P moves such that the angle APB is always 45°. What is the locus of P? Answer: The locus consists of two arcs of circles that have AB as a chord and subtend a 45° angle at any point on the arcs.
- Question: Sketch the locus of a point that is always 6 cm away from a fixed line L. Answer: Two lines parallel to L, one on each side, at a distance of 6 cm, form the locus.