Question

which of the following statements must be true based on the diagram below? select all that apply. (diagram is not to scale.) answer attempt 1 out of 2 (overline{mn}) is a segment bisector. (overline{mn}) is a perpendicular bisector. (m) is the vertex of a right angle. (n) is the vertex of a right angle. (m) is the midpoint of a segment in the diagram. (n) is the midpoint of a segment in the diagram.

Explanation:

Step1: Analyze the mid - point concept

Since \(KN = NL\) (marked with two - tick marks) and \(JM=ML\) (marked with two - tick marks), by the definition of a mid - point (a point that divides a segment into two equal parts), \(N\) is the mid - point of \(KL\) and \(M\) is the mid - point of \(JL\).

Step2: Analyze the segment bisector concept

A segment bisector is a line, ray, or segment that divides a segment into two equal parts. In \(\triangle KJL\), since \(N\) is the mid - point of \(KL\) and \(M\) is the mid - point of \(JL\), \(\overline{MN}\) is a mid - segment of \(\triangle KJL\). A mid - segment of a triangle is a segment that connects the mid - points of two sides of a triangle. By the mid - segment theorem, \(\overline{MN}\) bisects \(KL\) and \(JL\) (in the sense of connecting mid - points), so \(\overline{MN}\) is a segment bisector.

Step3: Analyze the perpendicular bisector concept

There is no information in the diagram (no right - angle marks) to suggest that \(\overline{MN}\) is perpendicular to \(KL\) or \(JL\). So, we cannot conclude that \(\overline{MN}\) is a perpendicular bisector.

Step4: Analyze the right - angle vertex concept

There are no right - angle marks at vertices \(M\) or \(N\) in the given diagram. So, we cannot conclude that \(M\) or \(N\) is the vertex of a right - angle.

Answer:

\(\overline{MN}\) is a segment bisector, \(M\) is the midpoint of a segment in the diagram, \(N\) is the midpoint of a segment in the diagram.