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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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Explain why the following statement is incorrect:

“In a circle (or in congruent circles) containing two unequal chords, the longer chord corresponds to the greater major arc.”

To determine

To explain:

The reason the given statement is incorrect.

Explanation

Given:

“In a circle (or in congruent circles) containing two unequal chords, the longer chord corresponds to the greater major arc.”

To Prove:

To prove that the given statement is incorrect:

“In a circle (or in congruent circles) containing two unequal chords, the longer chord corresponds to the greater major arc.”

Assume two chords AB¯, CD¯ and mAB¯>mCD¯.

The major arc of chord AB¯ is mACDB and the major arc of chord CD¯ is mCABD.

To prove that mACDB is not greater than mCABD i.e. mACDB<mCABD.

Theorem Used:

In a circle (or in congruent circles) containing two unequal minor arcs, the greater minor arc corresponds to the longer of the chords related to these arcs.

Proof:

Assume the minor arc of chord AB¯ is greater than minor arc of chord CD¯ i.e. mAB>mCD.

Applying the theorem,

In a circle (or in congruent circles) containing two unequal minor arcs, the greater minor arc corresponds to the longer of the chords related to these arcs.

And since the minor arcs are unequal i.e. mAB>mCD and from the given figure,

The chord corresponding to minor arc mCD is CD¯ and the chord corresponding to minor arc mAB is AB¯, we conclude that mAB¯>mCD¯

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