Introduction to Loci
Locus (plural: loci) is the path traced by a moving point that satisfies certain given conditions. In geometry, we construct loci to visualize all possible positions that satisfy specific rules.
In this lesson, we will learn how to construct various types of loci under different conditions, including points at fixed distances, equidistant points, and relationships with lines.
Locus from a Fixed Point
Construction Steps
The locus of points at a fixed distance r from a fixed point O is a circle with center O and radius r.
Steps to Construct
- Mark the fixed point O on your paper
- Set your compass to the required distance r
- With center at O, draw a complete circle
- All points on this circle are exactly distance r from O
Why This Works:
By definition, all points on a circle are equidistant from its center. Therefore, any point P moving while maintaining a constant distance r from O must lie on this circle.
Locus Equidistant from Two Fixed Points
Perpendicular Bisector Construction
The locus of points equidistant from two fixed points A and B is the perpendicular bisector of the line segment AB.
Construction Steps
- Draw line segment AB connecting the two points
- With A as center, draw an arc with radius > ½AB
- With B as center, draw another arc with same radius
- The arcs intersect at points above and below AB
- Connect these intersection points to form the perpendicular bisector
Proof:
Any point on the perpendicular bisector is equidistant from A and B because:
- It forms two congruent right triangles with A and B
- The common side (bisector) and equal halves of AB make the hypotenuses equal
Locus Equidistant from Two Intersecting Lines
Angle Bisector Construction
The locus of points equidistant from two intersecting straight lines is the angle bisector of the angle formed by the lines.
Construction Steps
- Draw the two intersecting lines (AB and AC)
- From the point of intersection A, mark equal lengths on both lines
- From these points, draw arcs that intersect
- Draw a line from A through this intersection point - this is the angle bisector
Why It Works:
The angle bisector is the set of all points equidistant from both lines because:
- It divides the angle into two equal parts
- Any point on it forms congruent right triangles with the lines
- The distances (perpendicular) to both lines must be equal
Locus Equidistant from Two Parallel Lines
Parallel Line Construction
The locus of points equidistant from two parallel lines is another line parallel to both and midway between them.
Construction Steps
- Draw the two parallel lines (AB and CD)
- Construct a perpendicular line connecting AB and CD
- Mark the midpoint of this perpendicular line
- Through this midpoint, draw a line parallel to AB and CD
Verification:
To prove two lines are parallel using locus:
- Construct equal distance markers from both lines
- Connect these markers to form the midline
- Measure angles to confirm parallelism (corresponding angles equal)
Locating the Center of a Circle
Center Construction Method
The center of a circle can be found by constructing perpendicular bisectors of any two chords.
Construction Steps
- Draw any two chords on the circle (not parallel)
- Construct the perpendicular bisector of each chord
- The point where these bisectors intersect is the center
- Verify by measuring equal distances to multiple points on the circle
Why This Works:
The perpendicular bisector of a chord always passes through the center of the circle, so the intersection of two such bisectors must be the center.
Constructing a Regular Hexagon
Hexagon Construction Methods
Method 1: Within a Circle
- Draw a circle with given radius
- Without changing compass width, mark a point on the circumference
- From this point, draw an arc intersecting the circumference
- Repeat until six equally spaced points are marked
- Connect these points to form the hexagon
Method 2: Using Intersecting Circles
- Draw a circle with given side length as radius
- Mark any point on circumference as first vertex
- Using same radius, draw arc from this point to intersect circle
- Repeat process around the circle
- Connect intersection points to form hexagon
Properties:
A regular hexagon has:
- All sides equal (same as circle radius)
- All internal angles equal to 120°
- Six lines of symmetry
Practice Exercise
Question 1
Construct the locus of points 4cm from a fixed point P.
This will be a circle with radius 4cm centered at point P.
Question 2
Construct the locus of points equidistant from points A and B that are 6cm apart.
This will be the perpendicular bisector of line segment AB.
Question 3
Construct the locus of points equidistant from two lines intersecting at 60°.
This will be the angle bisector, dividing the 60° angle into two 30° angles.
Question 4
Construct the locus of points equidistant from two parallel lines 8cm apart.
This will be a line parallel to both and exactly midway (4cm) between them.
Question 5
Given a circle with two chords, construct its center using perpendicular bisectors.
The intersection point of the perpendicular bisectors of any two chords is the center.
Question 6
Construct a regular hexagon with side length 5cm using the circle method.
Draw a circle with radius 5cm and mark six points around the circumference spaced at 60° intervals.
Ready for more challenges?
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Timed Test in 7 Levels
Test your loci construction skills through progressively challenging levels