**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°.

**Read More:**

- Construction of an Equilateral Triangle
- Construction Of Similar Triangle As Per Given Scale Factor
- Construction Of A Line Segment
- Construction Of The Bisector Of A Given Angle
- Construction Of An Angle Using Compass And Ruler

**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.

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