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A perpendicular is a straight line that is at a right angle with a given line or makes an angle of $90^{\circ}$ with another line. Perpendicular lines are frequently used in geometry and have wide applications.
Let’s understand how to construct a perpendicular line to a given line with steps and examples.
How to Construct a Perpendicular Line?
The first step to constructing the line segment is that draw the line segment for the given measurement. Then mark the point on the line and place the compass on the given point. Now draw the arc across the given line on each side. Without adjusting the compass, draw another arc.
How to Construct a Perpendicular Line From a Point Lying on it?
You can construct a perpendicular line to a given line from a point lying on it. For this, we draw an arc on the line with the given point and as centre and then draw arcs from the drawn arc.
Steps to Construct a Perpendicular Line From a Point Lying on it
The following steps are used to construct a perpendicular line from a point lying on it.
Consider a line segment $\text{AB}$ with a point $\text{P}$ on it.
We want to construct a line perpendicular to line $\text{AB}$ passing through the point $\text{P}$.
Step 1: With $\text{P}$ as a centre and any suitable radius, draw an arc cutting line segment $\text{AB}$ at two distinct points.
Step 2: Now with $\text{X}$ as a centre and a suitable radius draw an arc on either side of the given line segment $\text{AB}$. Also, with $\text{Y}$ as a centre and the same radius as the previous, draw an arc on either side of the given line segment cutting the arc drawn through the point $\text{X}$ at $\text{M}$.
Step 3: Join the points $\text{M}$ and $\text{P}$ as shown, and the line segment $\text{MP}$ is the required perpendicular to $\text{AB}$ through the point $\text{P}$.
Let us see if the line segment $\text{MP}$ constructed is actually perpendicular to $\text{AB}$ or not.
Join $\text{M}$ to the points $\text{X}$ and $\text{Y}$ as shown in the figure below.
Consider $\triangle \text{MXP}$ and $\triangle \text{MYP}$
$\text{XP} = \text{YP}$ (Drawn through the arc of the same radius with P as centre)
$\text{XM} = \text{YM}$ (M is on the arc of the same radius with X and Y as centres respectively)
$\text{MP} = \text{MP}$ (Common side)
$\triangle \text{MXP} \cong \triangle \text{MYP}$ (By SSS rule)
Therefore, $\angle \text{XPM} = \angle \text{YPM}$ (Corresponding angles of congruent triangles)
But $\angle \text{XPY} = 180^{\circ}$ (Straight line)
Therefore, $\angle \text{XPM} = \angle \text{YPM} = 90^{\circ}$
Thus, the constructed line segment $\text{MP}$ is perpendicular to $\text{AB}$.
How to Construct a Perpendicular Line From an External Point?
Now, let’s understand how to construct a perpendicular line from a point external to a given line, i.e., to construct a perpendicular line from a point not lying on a given line.
Steps to Construct a Perpendicular Line From an External Point
The following steps are used to construct a perpendicular line from a point not lying on it.
Consider a line segment $\text{AB}$ with an external point $\text{P}$.
We want to construct a line perpendicular to line $\text{AB}$ passing through the point $\text{P}$.
Step 1: Draw two arcs with the same radii and with centre $\text{A}$, cutting line $\text{AB}$ and $\text{D}$ and $\text{E}$ respectively.
Step 2: With $\text{D}$ and $\text{E}$ as centres and the same radius, draw arcs on the opposite side of the line cutting each other at $\text{M}$.
Step 3: Join $\text{P}$ and $\text{M}$.
Line $\text{PM}$ is perpendicular to the given line $\text{AB}$ from an external point $\text{P}$.
Practice Questions
- Draw a line segment of length 6 cm. Take a point P at a distance of 4 cm from the left end. Draw a perpendicular to the line passing through the point P.
- Draw a line segment of 8 cm. Take any point P not lying on the line. Draw a perpendicular to the line passing through the point P.
FAQs
What is the rule for perpendicular lines?
If two non-vertical lines in the same plane intersect at a right angle then they are said to be perpendicular. Horizontal and vertical lines are perpendicular to each other, for example, the axes of the coordinate plane.
Is perpendicular always $90^{\circ}$?
Yes, perpendicular lines are those lines that intersect each other at $90^{\circ}$.
Does perpendicular mean $180^{\circ}$?
No, perpendicular doesn’t mean $180^{\circ}$. $180^{\circ}$ forms a straight line, whereas perpendicular lines are at $90^{\circ}$.
Conclusion
A perpendicular is a straight line that is at a right angle with a given line or makes an angle of $90^{\circ}$ with another line. You can construct a line perpendicular to the given line using a ruler and a compass.
Recommended Reading
- How to Construct Parallel Lines(With Steps & Examples)
- How To Construct a Line Segment(With Steps & Examples)
- What are Collinear Points in Geometry – Definition, Properties & Examples
- What is a Transversal Line in Geometry – Definition, Properties & Examples
- What are Parallel Lines in Geometry – Definition, Properties & Examples
- What is Concurrent lines in Geometry – Definition, Conditions & Examples
- What is Half Line in Geometry – Definition, Properties & Examples
- What is a Perpendicular Line in Geometry – Definition, Properties & Examples
- Difference Between Axiom, Postulate and Theorem
- Lines in Geometry(Definition, Types & Examples)
- What Are 2D Shapes – Names, Definitions & Properties
- 3D Shapes – Definition, Properties & Types
