Bisecting a Line Segment
To bisect a line segment means to divide it into two equal halves. The point where it is divided is called the midpoint.
The construction you will learn draws a perpendicular bisector — a line that is perpendicular to the segment AND passes through its midpoint.
This construction uses only a compass and ruler. It is one of the most important constructions in geometry.
What is Bisecting a Line Segment - Grade 6 Maths (Practical Geometry)?
Definition: The perpendicular bisector of a line segment is a line that:
- Passes through the midpoint of the segment.
- Is perpendicular (at 90°) to the segment.
Key property: Every point on the perpendicular bisector is equidistant (at equal distance) from both endpoints of the segment.
Bisecting a Line Segment Formula
Steps to construct the perpendicular bisector of AB:
- Draw line segment AB.
- With A as centre, open compass to more than half of AB. Draw arcs above and below the line.
- With B as centre and the same radius, draw arcs above and below the line.
- The arcs above meet at point P. The arcs below meet at point Q.
- Join PQ. This line is the perpendicular bisector of AB.
- PQ crosses AB at point M. M is the midpoint of AB (AM = MB).
Why must the radius be more than half AB?
- If the radius is less than half, the arcs from A and B will not reach each other — they will not intersect.
- If the radius is exactly half, the arcs touch at one point (on the line), which is not useful.
- The radius must be more than half so the arcs cross at two clear points (P and Q).
Types and Properties
What can you do with a perpendicular bisector?
- Find the midpoint of any line segment without measuring.
- Check equal distances: Any point on the perpendicular bisector is equally far from A and B.
- Construct isosceles triangles: Pick any point on the bisector — it forms an isosceles triangle with A and B.
- Find the circumcentre: The perpendicular bisectors of all three sides of a triangle meet at one point (circumcentre).
Solved Examples
Example 1: Bisecting a 6 cm Segment
Problem: Draw AB = 6 cm. Construct its perpendicular bisector and find the midpoint.
Solution:
- Draw AB = 6 cm.
- Compass radius > 3 cm (say 4 cm). From A, draw arcs above and below.
- From B, same radius, draw arcs above and below. They meet at P (above) and Q (below).
- Join PQ. It cuts AB at M.
Answer: M is the midpoint. AM = MB = 3 cm.
Example 2: Bisecting an 8 cm Segment
Problem: Construct the perpendicular bisector of PQ = 8 cm.
Solution:
- Draw PQ = 8 cm.
- Compass radius > 4 cm (say 5 cm). Arcs from P and Q above and below.
- Join intersection points. The bisector cuts PQ at its midpoint (4 cm from each end).
Answer: Midpoint is at 4 cm from each end.
Example 3: Verifying with a Ruler
Problem: After bisecting AB = 10 cm, verify the midpoint M by measurement.
Solution:
Measure AM and MB with a ruler.
AM should be 5 cm. MB should be 5 cm.
If both are 5 cm, the construction is correct.
Example 4: Verifying the Perpendicular
Problem: After constructing the bisector of AB, verify that it is perpendicular.
Solution:
Use a protractor to measure the angle between the bisector and AB at point M.
It should be 90°.
Example 5: Using Perpendicular Bisector for Equal Distances
Problem: A and B are 8 cm apart. Find a point that is equally far from both A and B.
Solution:
- Draw AB = 8 cm.
- Construct the perpendicular bisector of AB.
- Any point on this bisector is equally far from A and B.
Answer: The midpoint M (4 cm from each) is one such point. Any point on the bisector works.
Example 6: Bisecting a 5.4 cm Segment
Problem: Find the midpoint of a 5.4 cm line segment by construction.
Solution:
- Draw AB = 5.4 cm.
- Compass > 2.7 cm. Draw arcs from A and B.
- Join intersection points. Bisector meets AB at M.
Answer: AM = MB = 2.7 cm.
Real-World Applications
Where perpendicular bisectors are used:
- Finding centres: The centre of a circle passing through two points lies on the perpendicular bisector of the segment joining them.
- Cricket/Football: The centre line of a field bisects the total length.
- Construction: Finding the exact middle of a beam or wall.
- Folding: When you fold a paper so two points coincide, the fold line is the perpendicular bisector.
- Navigation: Finding a point equidistant from two places.
Key Points to Remember
- The perpendicular bisector of a segment passes through its midpoint at 90°.
- Use a compass with radius more than half the segment length.
- Draw arcs from both endpoints with the same radius.
- Every point on the perpendicular bisector is equidistant from both endpoints.
- This construction finds the midpoint without measuring.
- Verify: measure both halves with a ruler, and check the angle is 90°.
Practice Problems
- Construct the perpendicular bisector of a 7 cm line segment.
- Draw AB = 9 cm and find its midpoint by construction.
- After bisecting a 12 cm segment, verify that each half is 6 cm.
- Construct the perpendicular bisector of a 4.6 cm segment.
- Mark a point on the perpendicular bisector of AB. Measure its distance from A and B. Are they equal?
- Construct a triangle with AB = 8 cm. Find the perpendicular bisector of AB.
Frequently Asked Questions
Q1. What does bisect mean?
Bisect means to divide into two equal parts. Bisecting a line segment means finding its midpoint.
Q2. Why must the compass radius be more than half the segment?
If the radius is too small, the arcs from the two endpoints will not reach each other. They need to intersect to create the points for the bisector.
Q3. Is the perpendicular bisector always perpendicular?
Yes, by construction. The line joining the two intersection points is always at exactly 90° to the original segment.
Q4. Can I find the midpoint without a perpendicular bisector?
You can measure with a ruler and divide by 2. But the compass construction is more accurate and is the standard geometric method.
Q5. What is the equidistant property?
Every point on the perpendicular bisector of AB is at equal distance from A and B. This is a key property used in many geometry problems.
Related Topics
- Constructing a Line Segment
- Constructing Perpendicular Lines
- Bisecting an Angle
- Constructing Angles
- Constructing Parallel Lines
- Constructing Triangles (SSS)
- Constructing Triangles (SAS)
- Constructing Triangles (ASA)
- Constructing Triangles (RHS)
- Constructing Quadrilaterals
- Constructing Special Quadrilaterals
- Triangle Construction Problems
