Given that segment \( AC \) is the perpendicular bisector of segment \( BD \) and they intersect at point \( E \), we can deduce the following properties:
- A perpendicular bisector divides a segment into two equal lengths.
- Therefore, since \( AC \) is the perpendicular bisector of \( BD \), it must hold that \( BE = ED \).
Now looking at the answers provided:
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It must be the same length as segment \( EA \): This is incorrect because \( EA \) is not guaranteed to be equal to \( BE \).
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It must be the same length as segment \( CD \): The relationship between \( BE \) and \( CD \) is not dictated by the information given.
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It must be the same length as segment \( DE \): This is correct because, by the property of the perpendicular bisector, \( BE = ED \).
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It must be the same length as segment \( DE \) (repeated): This option is also correct for the same reason as above.
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It must be the same length as segment \( AB \): This is not necessarily true as it is unrelated to the lengths of \( BE \).
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It must be the same length as segment \( A B \) (repeated): This is also not correct as it is unrelated.
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It must be the same length as segment \( BC \): This is incorrect for the same reason above.
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It must be the same length as segment \( B C \) (repeated): This is also incorrect for the same reason above.
Thus, the most accurate and correct answer from the options is:
It must be the same length as segment \( DE \) (the one referring to segment \( DE \)).
This answer is accurate because it directly utilizes the property of the perpendicular bisector, confirming that \( BE = ED \).
1 answer