# Construction of Perpendicular Line through a Point: Steps for Construction (For CBSE, ICSE, IAS, NET, NRA 2022)

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Lines which intersect each other at right angles are perpendicular to each other. In other words, if two lines are perpendicular to each other then, the angle made by them at the point of intersection will be 90^{o} and vice versa.

## Constructing a Perpendicular to a Line through an External Point

Given: A line segment AB and a point P lying outside the line segment AB

To construct: A line perpendicular to line AB passing through point P

### Steps for Construction

Step 1: With point P as center and any suitable radius draw an arc cutting line segment AB at two distinct points X and Y as shown in the following figure.

Step 2: Now with X as center and a suitable radius draw an arc on opposite side of the given point P. Also, with Y as center and same radius as previous, draw an arc on opposite side of the given point P cutting the arc drawn through point X at M as shown in the following figure.

Step 3: Join the points P and M. The line segment PM is the required perpendicular to the line segment AB and passing through external point P.

To check if the line segment constructed in the way as discussed is perpendicular to line segment AB, join the points M and P to X and Y as shown below:

S. No | Statement | Reason |

1. | Drawn through arc of same radius with P as center | |

2. | M is Drawn through the arc of same radius with X and Y as center respectively. | |

3. | SSS axiom of congruency (PM is common) | |

4. | Corresponding parts of the congruency triangle | |

5. | SAS axiom congruency (RP is common) | |

Corresponding parts of the congruent triangle | ||

6. | Both these angles form a linear pair. |

### Question 1

Construct a line perpendicular to PQ through the point R on PQ.

Solution:

Step 1: Stretch your compasses a short length. It can be any length. Put the sharp end of the compasses on point R and Mark two arcs S and T on the line PQ.

Step 2: Stretch the compasses a little more. Put the sharp end of the compasses on point S and make an arc with pencil end. Do the same on point T but make sure that the second arc intersects the first arc.

Step 3: Draw a line from point R to the point where the arcs intersect. This line is perpendicular to PQ and passes through the point R.