Question

problem 3
here are 2 points in the plane.

1. using only a straightedge, can you find points in the plane that are the same distance from points a and b? explain your reasoning.
2. using only a compass, can you find points in the plane that are the same distance from points a and b? explain your reasoning.

in this diagram, line segment cd is the perpendicular bisector of line segment ab. assume the conjecture that the set of points equidistant from a and b is the perpendicular bisector of ab is true. select all statements that must be true.

Explanation:

Step1: Analyze straight - edge case

A straightedge can only be used to draw straight lines. Since we don't have a way to measure distances precisely with just a straightedge, we cannot find points equidistant from A and B using only a straightedge.

Step2: Analyze compass case

A compass can be used to draw circles. Place the compass on point A and set its radius to the distance between A and B. Then place it on point B and draw another circle. The intersection points of these two circles are equidistant from A and B. Also, any point on the perpendicular bisector of AB (which can be constructed with a compass) is equidistant from A and B.

Answer:

1. No. A straightedge can only draw lines and cannot measure distances to find equidistant points.
2. Yes. Intersection points of circles centered at A and B with radius equal to distance between A and B, and points on the perpendicular bisector of AB (constructible with a compass) are equidistant from A and B.