Question

a true conditional statement is given. \if a point is on the perpendicular bisector of a line segment, then the point is equidistant from the line segment endpoints.\ select phrases to describe the inverse of the statement. if a point is dropdown then the point is dropdown. dropdown options: equidistant from a line segment’s endpoints, not equidistant from a line segment’s endpoints, not on the perpendicular bisector of a line segment,

Explanation:

To find the inverse of a conditional statement "If \( p \), then \( q \)", we negate both the hypothesis (\( p \)) and the conclusion (\( q \)). The original statement is "If a point is on the perpendicular bisector of a line segment (\( p \)), then the point is equidistant from the line segment’s endpoints (\( q \))".

• Negating \( p \): "a point is not on the perpendicular bisector of a line segment"
• Negating \( q \): "the point is not equidistant from a line segment’s endpoints"

Wait, no—wait, the inverse is formed by negating both the hypothesis and the conclusion. Wait, let's correct that. The inverse of "If \( p \), then \( q \)" is "If not \( p \), then not \( q \)". Wait, no, actually, the inverse is "If not \( p \), then not \( q \)"? Wait, no: the inverse of a conditional statement \( p \to q \) is \(
eg p \to
eg q \). Wait, but let's re - examine the original statement:

Original: \( p \): "a point is on the perpendicular bisector of a line segment", \( q \): "the point is equidistant from the line segment’s endpoints"

Inverse: \(
eg p \to
eg q \), which is "If a point is not on the perpendicular bisector of a line segment, then the point is not equidistant from a line segment’s endpoints". But wait, the first dropdown is for the hypothesis of the inverse, and the second is for the conclusion.

Wait, the first blank is after "If a point is", so we need the negation of \( p \) for the hypothesis of the inverse. The negation of \( p \) is "not on the perpendicular bisector of a line segment". Then the conclusion of the inverse is the negation of \( q \), which is "not equidistant from a line segment’s endpoints".

Wait, but let's check the options. The first dropdown options: "equidistant from a line segment’s endpoints", "not equidistant from a line segment’s endpoints", "not on the perpendicular bisector of a line segment". Wait, no, the first blank is the hypothesis of the inverse. The original hypothesis is "on the perpendicular bisector...", so the inverse hypothesis is "not on the perpendicular bisector...", so the first blank should be "not on the perpendicular bisector of a line segment", and the second blank (conclusion of inverse) is "not equidistant from a line segment’s endpoints". Wait, but let's confirm the definition of inverse of a conditional statement.

The inverse of a conditional statement "If \( P \), then \( Q \)" is "If not \( P \), then not \( Q \)". So:

• \( P \): "a point is on the perpendicular bisector of a line segment"
• \(

eg P \): "a point is not on the perpendicular bisector of a line segment"

• \( Q \): "the point is equidistant from the line segment’s endpoints"
• \(

eg Q \): "the point is not equidistant from the line segment’s endpoints"

So the inverse statement is: "If a point is not on the perpendicular bisector of a line segment, then the point is not equidistant from a line segment’s endpoints."

So for the first blank (after "If a point is"), we choose "not on the perpendicular bisector of a line segment", and for the second blank (after "then the point is"), we choose "not equidistant from a line segment’s endpoints".

Answer:

First blank: not on the perpendicular bisector of a line segment; Second blank: not equidistant from a line segment’s endpoints