Comparing Line Segments AF and ED in a Square: Geometric Analysis

Segment Comparison with Diagonal Intersections

Look at the square below:

AAABBBDDDCCCEEEFFF

Is AF equal to ED?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:05 Is side A F equal to side E D? Let's find out!
00:09 We aren't sure if these lines are parallel, which is needed to form a parallelogram.
00:15 If the lines were parallel, then A F would indeed equal E D.
00:24 But, we don't have enough information to decide.
00:27 And there you have it! That's our solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Look at the square below:

AAABBBDDDCCCEEEFFF

Is AF equal to ED?

2

Step-by-step solution

Since it is not given that FD is parallel to AE, it cannot be argued that AF is necessarily equal to ED

3

Final Answer

No

Key Points to Remember

Essential concepts to master this topic
  • Positioning: Points on sides don't guarantee equal segment lengths
  • Technique: Calculate using coordinates: A(0,0), F(3/4s,0), E(1/4s,s), D(s,s)
  • Check: AF = 3s/4 while ED = 3s/4√2, so AF ≠ ED ✓

Common Mistakes

Avoid these frequent errors
  • Assuming segments look equal means they are equal
    Don't assume AF = ED just because they appear similar in the diagram = wrong geometric reasoning! Visual similarity doesn't guarantee mathematical equality. Always calculate actual lengths using coordinates or apply geometric theorems.

Practice Quiz

Test your knowledge with interactive questions

Look at the square below:

Is a parallelogram a square?

FAQ

Everything you need to know about this question

Why aren't AF and ED equal if they look the same length?

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Visual appearance can be deceiving! AF is horizontal along the top side, while ED is diagonal. Even though F and E might divide their respective sides proportionally, the directions of the segments are different, making their lengths unequal.

How can I prove they're not equal without coordinates?

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Look at the directions: AF is purely horizontal, while ED goes diagonally from a point on the bottom side to a corner. Since they're not parallel and don't have the same orientation, they can't be equal unless proven by specific geometric conditions.

What if F and E are midpoints of their sides?

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Even with midpoints, AF would be s2 \frac{s}{2} (half the side), but ED would be s22 \frac{s\sqrt{2}}{2} (diagonal distance). The diagonal nature of ED makes it longer than the horizontal AF.

When would these segments be equal?

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They would only be equal under very specific conditions that aren't given in this problem. Without additional constraints like parallel lines or special angle relationships, we cannot assume equality.

How do I avoid making assumptions in geometry?

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  • Only use given information
  • Don't assume from visual appearance
  • Calculate when possible
  • Apply proven theorems, not hunches

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Properties, Characteristics and proofs

Find Diagonal AC: Square and Parallelogram with Area 100
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