Question

in $\triangle jkl$, $\overline{jo}=44$ in.
what is the length of $\overline{op}$?
46
52
22

Explanation:

Step1: Identify Medians and Centroid

In a triangle, the medians intersect at the centroid, which divides each median into a ratio of \(2:1\) (the segment from the vertex to the centroid is twice the segment from the centroid to the midpoint of the side). Here, \(M\) and \(N\) are midpoints (marked by congruent segments), so \(JL\) and \(JK\) have midpoints \(M\) and \(N\), making \(LN\) and \(KM\) medians. The intersection \(O\) is the centroid. Wait, actually, looking at the segments: \(M\) is midpoint of \(JL\) (three marks), \(N\) is midpoint of \(JK\) (two marks), and \(P\) is midpoint of \(KL\) (one mark). So \(JO\) is part of median \(JL\)? Wait, no, \(JO\) and \(OP\): Wait, maybe \(JL\) has midpoint \(M\), \(JK\) midpoint \(N\), \(KL\) midpoint \(P\). Then the medians are \(LM\), \(KN\), and \(JP\)? Wait, no, the diagram: \(J\) to \(L\) has midpoint \(M\) (three marks), \(J\) to \(K\) has midpoint \(N\) (two marks), \(K\) to \(L\) has midpoint \(P\) (one mark). Then the medians are \(LN\) (from \(L\) to \(N\), midpoint of \(JK\)), \(KM\) (from \(K\) to \(M\), midpoint of \(JL\)), and \(JP\) (from \(J\) to \(P\), midpoint of \(KL\))? Wait, no, the intersection \(O\) is the centroid. Wait, the key is: in a triangle, the centroid divides each median into a ratio of \(2:1\), with the longer segment being from the vertex to the centroid. But wait, maybe \(JO\) and \(OP\): Wait, maybe \(JL\) is a side, and \(O\) is the centroid, so \(JO\) is twice \(OP\)? Wait, no, let's re-examine. Wait, the problem: \(JO = 44\) in. Wait, maybe \(JL\) has midpoint \(M\), \(KL\) has midpoint \(P\), so \(JP\) is a median? No, maybe \(JO\) is from \(J\) to centroid \(O\), and \(OP\) is from centroid \(O\) to midpoint \(P\) of \(KL\). Wait, no, the centroid divides the median into \(2:1\), so if \(JO\) is the longer part (from vertex \(J\) to centroid \(O\)), then \(OP\) would be half of \(JO\)? Wait, no: centroid divides median into \(2:1\), so vertex to centroid is \(2\) parts, centroid to midpoint is \(1\) part. So if \(JO\) is the \(2\) part, then \(OP\) is the \(1\) part. So \(OP=\frac{JO}{2}\).

Step2: Calculate \(OP\)

Given \(JO = 44\) in, then \(OP=\frac{44}{2}=22\) in.

Answer:

22 (corresponding to the option "22")