Question

given that point c is the circumcenter of △lmn, which of the following is true? cr = lc cr = cq two of these rn = lr

Explanation:

Step1: Recall circum - center property

The circum - center is equidistant from vertices.

Step2: Analyze \(CR = LC\)

\(CR\) is a distance from circum - center to a point on a side - bisector, \(LC\) is to a vertex. No equality.

Step3: Analyze \(CR = CQ\)

No information to suggest they are equal as \(Q\) and \(R\) are points on side - bisectors.

Step4: Analyze \(RN = LR\)

No information about triangle symmetry to suggest this equality.

Answer:

None of the above (assuming no correct option among the given ones based on the properties of the circum - center). If we consider the properties of the circum - center of a triangle:

1. The circum - center \(C\) of \(\triangle LMN\) is the point of intersection of the perpendicular bisectors of the sides of the triangle. The circum - center is equidistant from the vertices of the triangle, i.e., \(LC = MC=NC\). There is no reason for \(CR = LC\) in general as \(R\) is just a point on a side - bisector and not a vertex. Also, there is no information to suggest \(CR = CQ\) ( \(Q\) is a point on a side - bisector too) and without further information about the triangle being isosceles or equilateral, we cannot say \(RN = LR\).