Perpendicular Lines
Stand straight and stretch your arms sideways. Your body and your arms form a right angle (90°). This is the idea behind perpendicular lines — two lines that meet at a right angle.
Perpendicular lines are everywhere: the corner of a book, the edges of a door frame, the junction of a wall and the floor, the plus (+) sign, and the letter T. Wherever two lines cross at exactly 90°, they are perpendicular.
In NCERT Class 6 Mathematics, perpendicular lines are studied in the chapter Understanding Elementary Shapes. You will learn how to identify perpendicular lines, use the symbol ⊥, and understand the perpendicular bisector.
Knowing what perpendicular means is essential for geometry — it is used in defining rectangles, squares, altitudes of triangles, and coordinate axes.
What is Perpendicular Lines?
Definition: Two lines (or line segments or rays) are called perpendicular if they intersect at a right angle (90°).
Symbol:
AB ⊥ CD means "line AB is perpendicular to line CD"
Key terms:
- Right angle: An angle of exactly 90°. It is marked with a small square at the corner.
- Perpendicular lines: Two lines that form a 90° angle at their point of intersection.
- Perpendicular bisector: A line that is perpendicular to a given line segment AND passes through its midpoint (divides it into two equal halves).
- Foot of the perpendicular: The point where the perpendicular line meets the other line.
Important:
- Perpendicular lines form four right angles at the point of intersection.
- If two lines are perpendicular, each one is perpendicular to the other (the relationship is mutual).
- Perpendicular lines must intersect. If two lines do not meet, they cannot be perpendicular (they might be parallel).
- The symbol ⊥ is read as "is perpendicular to."
Perpendicular Lines Formula
Key facts about perpendicular lines:
If AB ⊥ CD, then the angle between AB and CD = 90°
At the point of intersection:
- All four angles formed are 90° each.
- The four angles add up to 360° (90 × 4 = 360).
Perpendicular bisector properties:
- Divides the line segment into two equal halves.
- Meets the line segment at 90°.
- Every point on the perpendicular bisector is equidistant from both endpoints of the line segment.
Testing for perpendicularity:
- Using a protractor: measure the angle — if it is 90°, the lines are perpendicular.
- Using a set square: place the right-angle corner at the intersection — if both edges align with the two lines, they are perpendicular.
Types and Properties
Types and examples of perpendicularity:
- Two full lines that intersect at 90°.
- They extend infinitely in both directions.
2. Perpendicular line segments:
- Two line segments that meet at 90°.
- Example: Two sides of a rectangle at a corner.
3. Perpendicular rays:
- Two rays from the same starting point that form a 90° angle.
- A special perpendicular that also bisects (cuts in half) the line segment.
- Example: The perpendicular bisector of AB passes through the midpoint of AB at 90°.
5. Perpendicular from a point to a line:
- The shortest line segment from an external point to a line.
- This is always perpendicular to the line.
- The length of this segment is called the distance from the point to the line.
6. Not perpendicular:
- Two lines that intersect at any angle other than 90° are NOT perpendicular.
- Parallel lines never intersect, so they are NOT perpendicular.
Solved Examples
Example 1: Example 1: Identifying perpendicular lines
Problem: Look at the letter L. Are the two strokes of L perpendicular?
Solution:
- The letter L has a vertical stroke and a horizontal stroke.
- They meet at the bottom-left corner.
- The angle between them is 90°.
Answer: Yes, the two strokes of the letter L are perpendicular.
Example 2: Example 2: Perpendicular lines in everyday objects
Problem: Name five everyday objects where perpendicular lines can be seen.
Solution:
- Corners of a book: The two edges of a book at any corner meet at 90°.
- Window frame: The horizontal and vertical bars are perpendicular.
- Plus sign (+): The horizontal and vertical lines cross at 90°.
- Tiles on the floor: The edges of square tiles meet perpendicularly.
- The letter T: The horizontal bar and vertical bar meet at 90°.
Answer: Book corner, window frame, plus sign, floor tiles, letter T.
Example 3: Example 3: How many right angles?
Problem: Two perpendicular lines meet at point O. How many right angles are formed?
Solution:
- Two intersecting lines always form 4 angles.
- If the lines are perpendicular, each of these 4 angles is a right angle (90°).
- Total: 4 right angles.
Verification: 4 × 90° = 360° (complete angle) ✓
Answer: 4 right angles are formed.
Example 4: Example 4: Checking with a protractor
Problem: Two lines meet at an angle of 87°. Are they perpendicular?
Solution:
- Perpendicular lines must meet at exactly 90°.
- 87° ≠ 90°.
Answer: No, they are NOT perpendicular. The angle must be exactly 90°.
Example 5: Example 5: Perpendicular bisector
Problem: A line segment AB is 8 cm long. Its perpendicular bisector is the line m. Where does m cross AB, and at what angle?
Solution:
- A perpendicular bisector bisects the line segment → crosses at the midpoint.
- Midpoint of AB = 8 ÷ 2 = 4 cm from each end.
- The perpendicular bisector is perpendicular → angle = 90°.
Answer: m crosses AB at its midpoint (4 cm from each end) at an angle of 90°.
Example 6: Example 6: Perpendicular sides of a rectangle
Problem: In rectangle ABCD, name all pairs of perpendicular sides.
Solution:
In a rectangle, all angles are 90°. So adjacent sides are perpendicular:
- AB ⊥ BC (at corner B)
- BC ⊥ CD (at corner C)
- CD ⊥ DA (at corner D)
- DA ⊥ AB (at corner A)
Answer: There are 4 pairs of perpendicular sides: AB⊥BC, BC⊥CD, CD⊥DA, DA⊥AB.
Example 7: Example 7: Shortest distance from a point to a line
Problem: A point P is outside a line l. Three segments are drawn from P to line l: PA = 7 cm, PB = 5 cm (perpendicular), and PC = 6 cm. Which is the shortest?
Solution:
- The perpendicular from a point to a line is the shortest distance.
- PB is perpendicular to l, so PB is the shortest.
- PB = 5 cm < PC = 6 cm < PA = 7 cm.
Answer: PB = 5 cm is the shortest. The perpendicular distance is always the shortest distance from a point to a line.
Example 8: Example 8: Perpendicular lines in coordinate axes
Problem: Are the x-axis and y-axis perpendicular?
Solution:
- The x-axis is horizontal.
- The y-axis is vertical.
- They meet at the origin (0, 0).
- The angle between them is 90°.
Answer: Yes. The x-axis and y-axis are perpendicular. This is why the coordinate system is also called the rectangular coordinate system.
Example 9: Example 9: Not perpendicular
Problem: Two lines cross each other making angles of 60° and 120°. Are they perpendicular?
Solution:
- The angles formed are 60°, 120°, 60°, 120° (opposite angles are equal).
- None of the angles is 90°.
Answer: No, they are NOT perpendicular. Perpendicular lines form angles of exactly 90°.
Example 10: Example 10: Clock hands
Problem: At what times do the clock hands form perpendicular lines?
Solution:
The hands are perpendicular when the angle between them is 90°.
- At 3 o'clock: minute hand at 12, hour hand at 3 → angle = 90°.
- At 9 o'clock: minute hand at 12, hour hand at 9 → angle = 90°.
Note: These are the exact times. The hands also form 90° at approximately 12:16, 12:49, 1:22, 1:55, etc. during the day.
Answer: The most obvious times are 3 o'clock and 9 o'clock.
Real-World Applications
Real-world uses of perpendicular lines:
- Construction: Builders use perpendicular lines to ensure walls are straight and floors are level. A plumb line hangs perpendicular to the ground.
- Roads: T-junctions and crossroads often have roads meeting at right angles (perpendicular).
- Sports: The penalty area lines on a football field are perpendicular to the goal line. Cricket pitch lines are perpendicular to the boundary.
- Furniture: Table legs are perpendicular to the tabletop. Shelves are perpendicular to the wall.
- Maps and navigation: The x-axis and y-axis on maps and GPS systems are perpendicular, creating a grid.
- Art and design: Graph paper is made of perpendicular horizontal and vertical lines. Artists use perpendicular guidelines for balanced compositions.
Key Points to Remember
- Perpendicular lines meet at a right angle (90°).
- The symbol is ⊥ (read as "is perpendicular to").
- Two perpendicular lines form four right angles at their intersection.
- A perpendicular bisector is perpendicular to a line segment AND passes through its midpoint.
- The shortest distance from a point to a line is the perpendicular distance.
- Every point on the perpendicular bisector is equidistant from both endpoints of the segment.
- In a rectangle, all pairs of adjacent sides are perpendicular.
- The x-axis and y-axis are perpendicular to each other.
- If the angle between two lines is NOT 90°, they are not perpendicular (even if close to 90°).
- Perpendicular lines must intersect — parallel lines are never perpendicular.
Practice Problems
- Name three letters of the English alphabet that have perpendicular lines in them.
- Two lines meet at an angle of 90°. How many right angles are formed at the intersection?
- Draw a line segment of 6 cm. Construct its perpendicular bisector. What is the length of each half?
- In a square ABCD, name all pairs of perpendicular sides.
- A point P is 8 cm from a line l. What is the perpendicular distance from P to l? Can any other distance from P to l be shorter?
- Are the diagonals of a square perpendicular to each other? Verify by drawing.
- Give two examples of perpendicular lines from a classroom.
- If AB ⊥ CD at point O, and ∠AOC = 90°, what are the measures of ∠AOD, ∠BOC, and ∠BOD?
Frequently Asked Questions
Q1. What are perpendicular lines?
Perpendicular lines are two lines that intersect at a right angle (90°). The symbol for perpendicular is ⊥. For example, the edges of a book at a corner are perpendicular.
Q2. What symbol represents perpendicular?
The symbol ⊥ represents perpendicular. We write AB ⊥ CD to mean 'line AB is perpendicular to line CD.' The small square at the angle in a diagram also indicates a right angle.
Q3. How many right angles do perpendicular lines form?
Two perpendicular lines form 4 right angles at their point of intersection. Each of the four angles is exactly 90°.
Q4. What is a perpendicular bisector?
A perpendicular bisector of a line segment is a line that is perpendicular to the segment (meets it at 90°) and also bisects it (passes through its midpoint). Every point on the perpendicular bisector is equidistant from both endpoints.
Q5. Can parallel lines be perpendicular?
No. Parallel lines never intersect, so they cannot form any angle, let alone a 90° angle. Perpendicular lines must intersect.
Q6. What is the shortest distance from a point to a line?
The shortest distance from a point to a line is the perpendicular distance — the length of the line segment drawn from the point perpendicular to the line. Any other line segment from the point to the line will be longer.
Q7. Are the sides of a rectangle perpendicular?
Yes. In a rectangle, every pair of adjacent sides is perpendicular. This is because all angles of a rectangle are 90°. A square, being a special rectangle, also has all adjacent sides perpendicular.
Q8. How can you check if two lines are perpendicular without a protractor?
Use a set square (the triangular ruler with a 90° corner). Place the 90° corner at the point of intersection. If both edges of the set square align with the two lines, the lines are perpendicular.
