Question

in \\( \triangle h i j, n \\) is the centroid. if \\( l j = 19 \\), find \\( i l \\).

Explanation:

Step1: Recall the property of the centroid

The centroid of a triangle divides each median into a ratio of \(2:1\). Also, the median of a triangle bisects the opposite side. So, if \(N\) is the centroid and \(IL\) is a median, then \(IL = IL + LJ\) and \(IL\) is composed of two - thirds and one - third parts. But more simply, since \(IL\) is a median, \(IL=IL + LJ\) and \(IL\) is twice \(LJ\) (because the median bisects the side). Wait, no, correct property: A median of a triangle divides the triangle into two triangles of equal area and the centroid divides the median such that the distance from the vertex to the centroid is twice the distance from the centroid to the mid - point of the side. But for the length of the median, if \(LJ\) is half of \(IJ\) (since \(IL\) is a median), \(IL = 2\times LJ\).

Since \(IL\) is a median of \(\triangle HIJ\) (because \(N\) is the centroid and centroids lie on medians), and by the definition of a median (a line segment joining a vertex to the mid - point of the opposite side), if \(LJ = 19\), then \(IL=IL + LJ\) and \(IL = 2\times LJ\).

Step2: Calculate \(IL\)

\(IL=2\times19 = 57\)

Answer:

\(IL = 57\)