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 both the true statements.
diagram of triangle abc with lines l, m, n (perpendicular bisectors), points a, b, c, d, j, k, l, e
point e is closer to point a than it is to point c.
point l is closer to point b than it is to point a.
point d is closer to point b than it is to point c.
point j is closer to point a than it is to point b or point c.
point k is closer to point c than it is to point a or point b.
point l is closer to point c than it is to point a or point b.

Explanation:

To solve this, we use the property of perpendicular bisectors: any point on the perpendicular bisector of a segment is equidistant from the segment’s endpoints. Also, for points inside the triangle, their distances to vertices relate to their position relative to the perpendicular bisectors.

Analyzing Each Statement:

1. Point E is closer to A than C:

E is inside the triangle. Since lines \( l, m, n \) are perpendicular bisectors, the circumcenter \( J \) (their intersection) is equidistant from \( A, B, C \). E is near the center of the triangle, so it is not necessarily closer to \( A \) than \( C \). False.

1. Point L is closer to B than A:

Point \( L \) lies on the perpendicular bisector of a side (e.g., between \( B \) and the bisector of \( AC \) or \( AB \)). By the perpendicular bisector property, if \( L \) is closer to \( B \) (visually, its position relative to \( B \) and \( A \) suggests it is nearer to \( B \)). True.

1. Point D is closer to B than C:

Point \( D \) is near \( A \) and \( J \). Since \( J \) is equidistant from \( B \) and \( C \), and \( D \) is above \( J \) (closer to \( A \)), it is not closer to \( B \) than \( C \). False.

1. Point J is closer to A than B or C:

\( J \) is the circumcenter (intersection of perpendicular bisectors), so \( JA = JB = JC \). Thus, \( J \) is equidistant from \( A, B, C \). False.

1. Point K is closer to C than A or B:

Point \( K \) lies near the perpendicular bisector of \( AC \) (or another side) and is closer to \( C \) (visually, its position relative to \( C \) and \( A/B \) suggests it is nearer to \( C \)). True.

1. Point L is closer to C than A or B:

From statement 2, \( L \) is closer to \( B \), so it cannot be closer to \( C \) than \( A \) or \( B \). False.

Answer:

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