Question

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

Explanation:

Step1: Recall circum - center property

The circum - center of a triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle.

Step2: Analyze side - bisecting property

Since point $C$ is the circum - center of $\triangle LMN$, the perpendiculars from $C$ to the sides of the triangle bisect the sides. $CR$ is perpendicular to $LN$. So, $LR=RN$ because the perpendicular from the circum - center to a side of the triangle bisects that side.

Step3: Analyze other options

For option B, there is no information given to suggest that $MQ = QN$. For option C, $CM$ is the distance from the circum - center to a vertex and $CR$ is the distance from the circum - center to the mid - point of a side, and in general $CM
eq CR$.

Answer:

D. $RN = LR$