How to Solve: Loci Problems

How to Solve: Loci Problems

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Imagine you’re designing a theme park ride—where should you place safety barriers so every seat stays exactly 5 meters from the center? That’s a loci problem, and mastering it means you’ll nail exam questions on paths, distances, and geometric constraints!

What You Need To Know First

Before diving into loci, ensure you understand: 1. Basic constructions (compass, ruler, protractor). 2. Distance from a point/line (perpendicular distance, radius). 3. Equation of a circle (center-radius form).

Key Vocabulary

Formulas To Know

Step-by-Step Method

Step 1: Read the condition carefully

• Underline the key phrase (e.g., "exactly 3 cm from point A," "equidistant from lines L and M").
• Circle any fixed points/lines.

Step 2: Translate the condition into a geometric rule

• Distance from a point? → Circle.
• Distance from a line? → Parallel lines (or a single line if "on the line").
• Equidistant from two points? → Perpendicular bisector.
• Equidistant from two lines? → Angle bisector(s).
• Region (e.g., "less than 5 cm")? → Shaded area inside/outside a circle.

Step 3: Sketch the fixed elements

• Draw the given points/lines lightly in pencil.
• Label them (e.g., "Point A," "Line L").

Step 4: Apply the rule to draw the locus

• Circle: Use a compass. Set radius to the given distance. Draw around the fixed point.
• Parallel lines: Measure the given distance from the fixed line. Draw two lines (one on each side).
• Perpendicular bisector: Find the midpoint of the two points. Draw a line at 90° to the segment joining them.
• Angle bisector: Use a protractor to split the angle into two equal parts. Draw the bisector.

Step 5: Check for hidden conditions

• Is the locus a line or a region? (e.g., "within 5 cm" = shaded circle).
• Are there multiple conditions? (e.g., "equidistant from A and B and 3 cm from C" = intersection of bisector and circle).

Step 6: Label the locus clearly

• Write the condition next to the locus (e.g., "All points 4 cm from A").
• If it’s a region, shade it and add a key.

Step 7: Verify with an example point

• Pick a point on your locus. Does it satisfy the condition? If not, redraw.

Worked Example Using the Steps

Problem: Draw the locus of points exactly 2 cm from a fixed line AB.

1. Condition: "Exactly 2 cm from line AB."
2. Rule: Parallel lines at 2 cm distance.
3. Sketch: Draw line AB.
4. Apply rule:
5. Measure 2 cm above AB. Draw a parallel line.
6. Measure 2 cm below AB. Draw another parallel line.
7. Hidden conditions: None (it’s a line, not a region).
8. Label: "Locus: 2 cm from AB."
9. Verify: Pick a point on the upper line. Its perpendicular distance to AB is 2 cm. ✔️

Worked Examples

Example 1 – Basic: Circle Locus

Problem: A point P moves so that it is always 3 cm from a fixed point O. Draw the locus of P.

Solution: 1. Condition: "3 cm from fixed point O." 2. Rule: Circle with center O, radius 3 cm. 3. Sketch: Mark point O. 4. Apply rule: Use a compass. Set radius to 3 cm. Draw circle around O. 5. Hidden conditions: None. 6. Label: "Locus: All points 3 cm from O." 7. Verify: Pick a point on the circle. Distance from O = 3 cm. ✔️

What we did and why: The set of all points at a fixed distance from a point is a circle. We used a compass to ensure accuracy.

Example 2 – Medium: Perpendicular Bisector

Problem: Points A and B are 6 cm apart. Draw the locus of points equidistant from A and B.

Solution: 1. Condition: "Equidistant from A and B." 2. Rule: Perpendicular bisector of AB. 3. Sketch: Draw segment AB (6 cm). 4. Apply rule:
- Find midpoint M of AB (3 cm from A and B).
- Draw a line through M at 90° to AB. 5. Hidden conditions: None. 6. Label: "Locus: Equidistant from A and B." 7. Verify: Pick a point on the bisector. Measure distances to A and B—they’re equal. ✔️

What we did and why: The perpendicular bisector is the set of all points equidistant from two fixed points. We constructed it using the midpoint and a right angle.

Example 3 – Exam Style: Combined Conditions

Problem: A point P moves so that it is equidistant from points X and Y and also 4 cm from point Z. X and Y are 8 cm apart. Z is 5 cm from X and 3 cm from Y. Draw the locus of P.

Solution: 1. Condition 1: "Equidistant from X and Y."
- Rule: Perpendicular bisector of XY.
- Sketch: Draw XY = 8 cm. Find midpoint M (4 cm from X and Y). Draw bisector. 2. Condition 2: "4 cm from Z."
- Rule: Circle with center Z, radius 4 cm.
- Sketch: Mark Z (5 cm from X, 3 cm from Y). Draw circle. 3. Hidden condition: P must satisfy both conditions → intersection points. 4. Apply rules:
- Draw the bisector.
- Draw the circle.
- Mark where they intersect (two points). 5. Label: "Locus: Points equidistant from X/Y and 4 cm from Z." 6. Verify: Pick an intersection point. Check distances to X/Y (equal) and to Z (4 cm). ✔️

What we did and why: This is a combined locus. We found the intersection of two loci (bisector and circle) to satisfy both conditions.

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, let’s lock this in. Loci problems are about finding all points that follow a rule. Here’s the cheat sheet: 1. Circle? Fixed distance from a point → compass, radius = distance. 2. Parallel lines? Fixed distance from a line → measure, draw two lines. 3. Perpendicular bisector? Equidistant from two points → midpoint + 90°. 4. Angle bisector? Equidistant from two lines → split the angle. 5. Combined conditions? Draw both loci, find where they cross. Double-check: Did you label it? Did you shade regions? Did you verify with a test point? Now go crush that exam—you’ve got this!