# How Do You Construct A Perpendicular Bisector

## Construction Of Perpendicular Bisector Of A Line Segment

A line which is perpendicular to a given line segment (AB) and divides it into two equal halves, i.e., AO = OB is called the perpendicular bisector of AB.
In figure, XY is the perpendicular bisector of AB since AO = OB and ∠XOB = 90°.

To draw a perpendicular bisector of a line segment
Construction: Draw the perpendicular bisector of a line segment AB = 5.5 cm using a scale and a pair of compasses.

• Step 1: Draw a line segment AB of length 5.5 cm.
• Step 2: Taking A as the centre and with any radius more than half of AB, draw an arc on both side of AB.
• Step 3: Similarly, taking B as the centre and radius as in step 2, draw another arc on both side of AB intersecting the previous arcs at C and D.
• Step 4: Join C and D crossing AB at O.
Hence, CD is the required perpendicular bisector of AB.

Verification: Measure AO and OB. We find the measurement of AO = OB and also ∠COB = ∠COA = 90°.

Example 1:    Draw a line segment PQ of length 8.4 cm. Draw the perpendicular bisector of this line segment.
Solution:    We follow the following steps for constructing the perpendicular bisector of PQ.
Steps of Construction:
Step I: Draw a line segment PQ = 8.4 cm by using a ruler.
Step II: With P as centre and radius more than half of PQ, draw two arcs, one on each side of PQ.
Step III: With Q as centre and the same radius as in step II, draw arcs cutting the arcs drawn in the previous step at L and M respectively.
Step IV: Draw the line segment with L and M as end-points.
The line segment LM is the required perpendicular bisector of PQ.

### To draw a perpendicular at a point  on the line

Construction: Draw a perpendicular at a point on the line segment AB = 5.5 cm using a scale and a
pair of compasses.
Given: A line segment AB of length 5.5 cm and a 1 point P lying on it.
To construct: A line passing through P being perpendicular to AB

• Step 1: Draw a line segment AB of length 5.5 cm and make a point P on it.
• Step 2: Taking P as the centre and with any convenient radius, draw an arc cutting AB at X and Y.
• Step 3: Taking X and Y as centres and with any suitable radius draw arcs cutting each other at Q.
• Step 4: Join P and Q.
Then PQ is perpendicular to AB passing through the point P.

### To draw a perpendicular to a given line from a point lying outside the line

Construction: Draw a perpendicular from a point outside the line segment AB = 5.5 cm.
Given: A line segment AB of length 5.5 cm and a point Y lying outside the line.
To construct: A line passing through Y which is perpendicular to AB.

• Step 1: Draw a line segment AB of length 5.5 cm and mark point Y outside the line segment AB.
• Step 2: Taking Y as the centre and with any suitable radius, draw an arc cutting AB at C and D.
• Step 3: Taking C and D as centres and with radius more than half of CD, draw arcs below AB intersecting each other at X.
• Step 4: Join X and Y.
Hence, XY is the required perpendicular to the line segment AB from point Y lying outside the line segment AB.