Question

the diagram is a straightedge and compass construction. lines l, m, and n are the perpendicular bisectors of the sides of triangle abc. select all the true statements. a point e is closer to point a than it is to point c. b point l is closer to point b than it is to point a. c point d is closer to point b than it is to point c. d point j is closer to point a than it is to point b or point c. e point k is closer to point c than it is to point a or point b. f point l is closer to point c than it is to point a or point b.

Explanation:

Step1: Recall property of perpendicular bisector

A point on the perpendicular bisector of a line - segment is equidistant from the endpoints of the line - segment. Points not on the perpendicular bisector are closer to one of the endpoints.

Step2: Analyze point E

There is no information about the position of point E with respect to the perpendicular bisectors to determine its distance from A and C. So, we cannot say if point E is closer to point A than to point C.

Step3: Analyze point L

Point L lies on the perpendicular bisector of side AB. So, point L is equidistant from A and B. So, the statement "Point L is closer to point B than it is to point A" is false.

Step4: Analyze point D

There is no information about the position of point D with respect to the perpendicular bisectors to determine its distance from B and C. So, we cannot say if point D is closer to point B than to point C.

Step5: Analyze point J

Point J is the circum - center (the point of intersection of the perpendicular bisectors of the sides of the triangle). So, JA = JB=JC. So, the statement "Point J is closer to point A than it is to point B or point C" is false.

Step6: Analyze point K

Point K lies in the region closer to side AC. Since it is on the side of the perpendicular bisector of AC that is closer to C, point K is closer to point C than it is to point A or point B.

Step7: Analyze point L again

As mentioned before, point L is on the perpendicular bisector of AB, so it is not closer to C than to A or B.

Answer:

E. Point K is closer to point C than it is to point A or point B.