Question

in \\( \triangle lmn \\), \\( r \\) is the centroid. if \\( mp = 14 \\), find \\( mn \\).

Explanation:

Step1: Recall the property of the centroid

The centroid of a triangle divides each median into a ratio of \(2:1\). Also, a median of a triangle connects a vertex to the mid - point of the opposite side.

Step2: Use the mid - point property

Since \(R\) is the centroid and \(MP\) is a part of the median \(MP\) (where \(P\) is the mid - point of \(LN\) because \(MP\) is a median). By the definition of a mid - point, if \(P\) is the mid - point of \(LN\), then \(MN = 2MP\).

Step3: Calculate \(MN\)

Given \(MP = 14\), substituting into the formula \(MN=2MP\), we get \(MN = 2\times14=28\).

Answer:

\(28\)