Circle Geometry: Proving AB=CD with Intersecting Lines

Chord Properties with Diameter Comparison

M is the center of the circle.

Perhaps AB=CD AB=CD

MMMAAABBBCCCDDDEEEFFFGGGHHH

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Is AB equal to CD?
00:03 CD is a diameter - a chord passing through the center of the circle
00:09 Any chord that is not a diameter is shorter than the circle's diameter
00:13 AB is a chord that is not a diameter, therefore it must be shorter than the diameter
00:16 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

M is the center of the circle.

Perhaps AB=CD AB=CD

MMMAAABBBCCCDDDEEEFFFGGGHHH

2

Step-by-step solution

CD is a diameter, since it passes through the center of the circle, meaning it is the longest segment in the circle.

AB does not pass through the center of the circle and is not a diameter, therefore it is necessarily shorter.

Therefore:

ABCD AB\ne CD

3

Final Answer

No

Key Points to Remember

Essential concepts to master this topic
  • Diameter Rule: Any line through the center is the longest chord
  • Recognition: CD passes through M, making it a diameter
  • Check: Compare chord positions: AB misses center, CD hits center ✓

Common Mistakes

Avoid these frequent errors
  • Assuming all chords in a circle have equal length
    Don't think AB = CD just because both are chords = ignoring the diameter property! The diameter is always the longest chord in any circle. Always check if a chord passes through the center to identify diameters.

Practice Quiz

Test your knowledge with interactive questions

There are only 4 radii in a circle.

FAQ

Everything you need to know about this question

How can I tell which chord is longer just by looking?

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Look at the center point M! Any chord that passes through the center is a diameter - the longest possible chord. Chords that miss the center are always shorter.

Why is the diameter always the longest chord?

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A diameter goes all the way across the circle through its widest part. Any other chord cuts across at a narrower section, making it shorter than the full width.

Can two different chords ever be equal in length?

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Yes! Non-diameter chords can be equal if they're the same distance from the center. But a diameter is always longer than any non-diameter chord.

What if I can't see the center clearly in the diagram?

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Look for the point labeled as the center (like M here), then trace which lines pass directly through it. Those lines extend to form diameters on the circle.

Does the direction of the chord matter for its length?

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No! Whether a chord goes horizontally, vertically, or diagonally doesn't affect its length. Only its distance from the center determines how long it is.

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