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Chapter 1: Chapter 1

Problem 10

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Write the inverse for each of the following statements. Determine whether the inverse is true or false, (a) If a person is stealing, he is breaking the law. (b) if a line is perpendicular to a segment at its midpoint, it is the perpendicular bisector of the segment, (c) Dead men tell no tales.

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(a) Inverse statement: If a person is not stealing, he is not breaking the law. This inverse is false. (b) Inverse statement: If a line is not perpendicular to a segment at its midpoint, it is not the perpendicular bisector of the segment. This inverse is true. (c) Inverse statement: Men who are not dead do tell tales. This inverse is true.
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TABLE OF CONTENTS :
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Step 1: Statement (a)

Original statement: If a person is stealing, he is breaking the law. To form the inverse, we negate both the hypothesis and the conclusion: Inverse statement: If a person is not stealing, he is not breaking the law. Now let's analyze the truth of the inverse: It is possible for someone to not steal and still break the law (e.g., speeding). Therefore, the inverse of statement (a) is false.

Step 2: Statement (b)

Original statement: If a line is perpendicular to a segment at its midpoint, it is the perpendicular bisector of the segment. To form the inverse, we negate both the hypothesis and the conclusion: Inverse statement: If a line is not perpendicular to a segment at its midpoint, it is not the perpendicular bisector of the segment. Now let's analyze the truth of the inverse: A line being the perpendicular bisector of the segment requires it to be both perpendicular to the segment and passing through its midpoint. If a line is not perpendicular or not passing through the midpoint, then it cannot be the perpendicular bisector. Therefore, the inverse of statement (b) is true.

Step 3: Statement (c)

Original statement: Dead men tell no tales. To form the inverse, we negate both the hypothesis and the conclusion: Inverse statement: Men who are not dead do tell tales. Now let's analyze the truth of the inverse: Men who are alive can tell stories or tales. Therefore, the inverse of statement (c) is true.

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