Constructing Perpendicular Lines
Two lines are perpendicular when they meet at a right angle (90°). The symbol for perpendicular is ⊥.
In Class 6 Practical Geometry, you will learn to construct perpendicular lines using a ruler and compass — without using a protractor.
There are two main constructions: (1) drawing a perpendicular to a line through a point on the line, and (2) drawing a perpendicular from a point outside the line.
What is Constructing Perpendicular Lines - Grade 6 Maths (Practical Geometry)?
Perpendicular from a point ON the line:
Given a line l and a point P on it, draw a line through P that is at 90° to l.
Perpendicular from a point OUTSIDE the line:
Given a line l and a point P not on it, draw a line from P to l that meets l at 90°.
Both constructions use only a ruler and compass.
Constructing Perpendicular Lines Formula
Construction 1: Perpendicular through a point on the line
- You have line l and point P on it.
- With P as centre and any radius, draw arcs on both sides of P on line l. Call the points where the arcs cut the line A and B.
- With A as centre and radius more than AP, draw an arc above the line.
- With B as centre and the same radius, draw another arc above the line. Let the two arcs meet at point Q.
- Join PQ. The line PQ is perpendicular to l at P.
Construction 2: Perpendicular from a point outside the line
- You have line l and point P above (or below) it.
- With P as centre and a radius large enough to cut line l, draw an arc that cuts line l at two points A and B.
- With A as centre and radius more than half of AB, draw an arc on the other side of l (opposite to P).
- With B as centre and the same radius, draw another arc. Let the arcs meet at Q.
- Join PQ. This line meets l at a point M. PM is perpendicular to l.
Types and Properties
Why are perpendicular constructions important?
- You need perpendiculars to construct squares and rectangles.
- Perpendicular bisectors are used to find midpoints and circumcentres.
- Altitudes of triangles are perpendicular lines from vertices to opposite sides.
Tips for accurate construction:
- Keep the compass radius the same when drawing arcs from A and B.
- Make the radius large enough so that the arcs clearly intersect.
- Use thin, clear arcs — not thick or smudged lines.
- Do not erase construction arcs — they show your method.
Solved Examples
Example 1: Perpendicular at a Point on a Line
Problem: Draw a line l. Mark a point P on it. Construct a perpendicular to l at P.
Solution:
- Draw line l. Mark point P on it.
- From P, draw arcs of equal radius on both sides → points A and B.
- From A and B, draw arcs with equal (larger) radius above l → they meet at Q.
- Join PQ.
Answer: PQ ⊥ l at P.
Example 2: Perpendicular from a Point Outside
Problem: Draw a line l. Mark a point P above it. Construct a perpendicular from P to l.
Solution:
- From P, draw an arc cutting l at A and B.
- From A and B, draw arcs below l with equal radius → they meet at Q.
- Join PQ. It meets l at M.
Answer: PM ⊥ l, with M on l.
Example 3: Constructing a Right Angle
Problem: At point A on a line, construct a 90° angle.
Solution:
This is the same as constructing a perpendicular at A. Follow Construction 1.
Answer: The angle between the line and the perpendicular is 90°.
Example 4: Perpendicular at the Endpoint of a Segment
Problem: Given line segment AB, construct a perpendicular at B.
Solution:
- Extend line AB beyond B if needed.
- Use Construction 1 with P = B on line AB.
Answer: The perpendicular at B makes 90° with AB.
Example 5: Checking with a Protractor
Problem: After constructing a perpendicular, how can you verify it is correct?
Solution:
Place a protractor at the meeting point. Measure the angle between the two lines. It should be 90°.
Example 6: Perpendicular at the Midpoint
Problem: Construct a perpendicular to line segment AB at its midpoint.
Solution:
- This is actually the perpendicular bisector of AB (covered in the bisecting lesson).
- Open compass to more than half of AB. From A, draw arcs above and below. From B, draw arcs with same radius. Connect the intersection points.
Answer: The line passes through the midpoint of AB and is perpendicular to it.
Real-World Applications
Where perpendicular constructions are used:
- Building construction: Walls must be perpendicular to the floor.
- Carpentry: Table legs must be at right angles to the tabletop.
- Drawing: Graph paper is made of perpendicular horizontal and vertical lines.
- Road design: Crossroads often meet at right angles.
- All rectangle/square constructions require perpendicular lines.
Key Points to Remember
- Perpendicular lines meet at 90°.
- Two constructions: perpendicular at a point on the line, and perpendicular from a point outside the line.
- Both use only a ruler and compass — no protractor needed.
- The arcs from two points must have the same radius to ensure accuracy.
- Do not erase construction arcs — they are part of the answer.
- Verify with a protractor: the angle should be 90°.
Practice Problems
- Draw a line and mark a point on it. Construct a perpendicular at that point.
- Draw a line and mark a point 3 cm above it. Construct a perpendicular from the point to the line.
- Construct a line segment AB = 6 cm. Construct a perpendicular at A.
- Construct a line segment PQ = 8 cm. Construct perpendiculars at both P and Q.
- Verify your perpendicular construction using a protractor.
- At the midpoint of a 10 cm segment, construct a perpendicular.
Frequently Asked Questions
Q1. What does perpendicular mean?
Two lines are perpendicular when they meet at exactly 90° (a right angle). The symbol is ⊥.
Q2. Can I use a protractor to draw a perpendicular?
Yes, you can measure 90° with a protractor. But in Practical Geometry, you are expected to use a compass and ruler for constructions.
Q3. Why must the arcs from A and B have equal radius?
Equal radii ensure the intersection point is equidistant from A and B. This guarantees the line through the intersection and the original point is exactly 90° to the line.
Q4. What if the arcs do not intersect?
The radius is too small. Open the compass wider — the radius from each point must be more than half the distance AB.
Q5. What is the foot of the perpendicular?
When you draw a perpendicular from a point P to a line l, the point where the perpendicular meets the line is called the foot of the perpendicular.
Related Topics
- Constructing a Line Segment
- Bisecting a Line Segment
- Perpendicular Lines
- Constructing Angles
- Bisecting an Angle
- Constructing Parallel Lines
- Constructing Triangles (SSS)
- Constructing Triangles (SAS)
- Constructing Triangles (ASA)
- Constructing Triangles (RHS)
- Constructing Quadrilaterals
- Constructing Special Quadrilaterals
- Triangle Construction Problems
