Question

select the term that matches the statement manipulated from the conditional below, then determine the validity of the statement: if a point divides a segment into two equal halves, then it is a midpoint. a. if a point does not divide a segment into two equal halves, then it is not a midpoint. this is the dropdown validity: dropdown b. if a point is not a midpoint, then it does not divide a segment into two equal halves. this is the dropdown validity: dropdown c. if a point is a midpoint, then it divides a segment into two equal halves. this is the dropdown validity: dropdown

Explanation:

1. Recall the definitions of conditional - related statements:

• The original statement is "If a point divides a segment into two equal halves, then it is a mid - point". Let \(p\) be "a point divides a segment into two equal halves" and \(q\) be "it is a mid - point". The original statement is \(p

ightarrow q\).

• For statement A, "If a point does not divide a segment into two equal halves, then it is not a mid - point". This is the contrapositive of the original statement. The contrapositive of \(p

ightarrow q\) is \(
eg q
ightarrow
eg p\). A conditional statement and its contrapositive are logically equivalent, so it is valid.

• For statement B, "If a point is not a mid - point, then it does not divide a segment into two equal halves". This is the inverse of the original statement. The inverse of \(p

ightarrow q\) is \(
eg p
ightarrow
eg q\). An inverse is not logically equivalent to the original conditional statement, but in the context of the definition of a mid - point, it is valid because by the definition, only mid - points divide a segment into two equal halves.

• For statement C, "If a point is a mid - point, then it divides a segment into two equal halves". This is the converse of the original statement. The converse of \(p

ightarrow q\) is \(q
ightarrow p\). A converse is not logically equivalent to the original conditional statement, but in the context of the definition of a mid - point, it is valid because the definition of a mid - point implies this relationship.

1. Identify the types of statements:

• Statement A is the contrapositive.
• Statement B is the inverse.
• Statement C is the converse.
• All of them are valid based on the definition of a mid - point.

Answer:

A. Contrapositive, Valid
B. Inverse, Valid
C. Converse, Valid