Counting Perpendicular Lines: Finding All Line Pairs in a Square

Perpendicular Lines with Square Corner Analysis

How many pairs of perpendicular lines are there in the square below?

AAABBBCCCDDD

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 How many perpendiculars are there in the drawing?
00:03 All angles in the square are right angles
00:07 A perpendicular creates a right angle at the intersection point between the lines
00:13 Let's mark and count all the right angles
00:19 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

How many pairs of perpendicular lines are there in the square below?

AAABBBCCCDDD

2

Step-by-step solution

Let's remember that perpendicular lines are lines that form a right angle of 90 degrees between them.

Our right angles are:

ABC,BCD,CDA,DAB ABC,BCD,CDA,DAB

The lines that form angle ABC are: AB+BC

The lines that form angle BCD are: BC+CD

The lines that form angle CDA are: CD+DA

The lines that form angle DAB are: DA+AB

Since in a rectangle we have 4 marked angles of 90 degrees, we must have 4 pairs of perpendicular lines.

3

Final Answer

4

Key Points to Remember

Essential concepts to master this topic
  • Definition: Perpendicular lines meet at exactly 90-degree angles
  • Technique: Count each corner's two intersecting lines as one pair
  • Check: Square has 4 corners × 1 pair each = 4 pairs ✓

Common Mistakes

Avoid these frequent errors
  • Counting individual lines instead of pairs
    Don't count AB, BC, CD, DA as 4 separate items = wrong count! This misses that perpendicularity requires TWO lines working together. Always identify which two specific lines meet at each 90° angle to form one perpendicular pair.

Practice Quiz

Test your knowledge with interactive questions

What do the four figures below have in common?

1234

FAQ

Everything you need to know about this question

Why does each corner only count as one pair?

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Each corner has exactly two lines meeting at 90°. For example, corner A has lines AB and AD - that's one pair of perpendicular lines, not two separate items.

Are all the angles in a square really 90 degrees?

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Yes! By definition, a square has four right angles. Each corner shows two sides meeting perpendicularly, which is what creates those perfect 90° angles.

What if I accidentally counted some pairs twice?

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This happens when students count both AB ⊥ BC and BC ⊥ AB. Remember: order doesn't matter - these represent the same perpendicular relationship!

How do I identify perpendicular lines in other shapes?

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Look for the right angle symbols (small squares) at corners, or measure angles. If two lines meet at exactly 90°, they're perpendicular.

Could there be more than 4 pairs if we drew diagonals?

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Great thinking! But this problem only shows the four sides of the square. Diagonals would create additional perpendicular relationships, but they're not drawn here.

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