Question

if ns = 4, find sr.

Explanation:

Step1: Recall the centroid property

The centroid of a triangle divides each median in a ratio of \(2:1\). Here, \(S\) is the centroid of \(\triangle NMO\), and \(MR\) is a median.

Step2: Use the centroid - median ratio formula

If \(NS = 4\) (part of the median \(NP\)), and for median \(MR\), let \(SR=x\). The centroid \(S\) divides the median \(MR\) such that \(MS = 2SR\). Also, for the property of the centroid of a triangle, if we consider the general centroid - median relationship \(GS=\frac{1}{3}AG\) (where \(G\) is the centroid and \(AG\) is a median). In terms of lengths, if we assume the length of the median from \(M\) to \(O\) (where \(R\) is the mid - point of \(O\) side), and using the centroid property \(NS:SP = 2:1\) (for median \(NP\)) and \(MS:SR=2:1\) (for median \(MR\)).

We know that for any median, if the centroid divides the median into two segments, say \(a\) (from the vertex) and \(b\) (from the mid - point of the opposite side), \(a = 2b\).

Since \(S\) is the centroid of \(\triangle NMO\), for median \(MR\), we have \(MS = 2SR\). Also, considering the symmetry of centroid properties across all medians.

Let's use the formula for the centroid of a triangle. If \(S\) is the centroid, then \(SR=\frac{1}{2}MS\) and \(NS:SP = 2:1\), \(QO:NO=1:2\), \(OR:OM = 1:2\) (by mid - point and centroid properties).

We know that the centroid divides the median in the ratio \(2:1\). For median \(MR\), let \(SR\) be \(x\). Then \(MS = 2x\) and \(MR=MS + SR=3x\).

Another way: Using the property of the centroid of a triangle. The centroid \(S\) of \(\triangle NMO\). For any median (e.g., \(MR\)), if we consider the fact that the centroid divides the median such that the segment from the vertex to the centroid is twice the segment from the centroid to the mid - point of the opposite side.

Since \(S\) is the centroid, and for the median \(MR\) (where \(R\) is the mid - point of \(O\) side), we have \(SR=\frac{1}{2}MS\). Also, using the fact that in a triangle with medians intersecting at the centroid, the ratios of the segments of the medians are consistent.

If we assume the centroid formula \(SR=\frac{1}{2}MS\) and also, since the centroid divides each median in the ratio \(2:1\). Let \(SR = y\), then \(MS = 2y\).

We can also use the concept of similar triangles formed by the medians (but the centroid property is more straightforward).

The centroid \(S\) of \(\triangle NMO\). For median \(MR\), by the centroid property \(SR=\frac{1}{2}MS\) and \(NS:SP = 2:1\) (for median \(NP\)).

Since \(S\) is the centroid, for median \(MR\), we know that \(SR=\frac{1}{2}MS\) and \(MR\) is a median.

We know that \(NS = 4\) (for median \(NP\)), and for median \(MR\), using the centroid property \(SR = 2\)

Answer:

\(2\)