Question

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the medians of $\triangle def$ are $\overline{dk}$, $\overline{el}$, and $\overline{fj}$. they meet at a single point $m$. (in other words, $m$ is the centroid of $\triangle def$.) suppose $el = 24$, $dm = 12$, and $fm = 6$. find the following lengths. note that the figure is not drawn to scale. figure of triangle def with medians dk, el, fj intersecting at m $dk = \square$ $mj = \square$ $ml = \square$

Explanation:

Step1: Recall centroid properties

The centroid of a triangle divides each median into a ratio of \(2:1\), with the longer segment being closer to the vertex. So, for any median, the length from the vertex to the centroid is twice the length from the centroid to the midpoint of the side.

Step2: Find \(ML\)

Given \(EL = 24\), and \(M\) is the centroid. So, \(EM:ML=2:1\) and \(EL = EM + ML\). Let \(ML=x\), then \(EM = 2x\). So, \(2x + x=24\), \(3x = 24\), \(x=\frac{24}{3}=8\). So \(ML = 8\).

Step3: Find \(MJ\)

Given \(FM = 6\), and \(F J\) is a median (since \(M\) is centroid, \(F J\) is a median). So, \(FM:MJ = 2:1\). Let \(MJ = y\), then \(FM = 2y\). So, \(2y=6\), \(y = \frac{6}{2}=3\). Wait, no, wait: Wait, \(FM\) is from vertex \(F\) to centroid \(M\), so \(FM:MJ = 2:1\), so \(MJ=\frac{1}{2}FM\)? Wait, no, centroid divides the median into \(2:1\) (vertex to centroid : centroid to midpoint). So if \(FM\) is the segment from \(F\) (vertex) to \(M\) (centroid), then \(MJ\) is from \(M\) (centroid) to \(J\) (midpoint of \(DE\)). So \(FM:MJ = 2:1\), so \(MJ=\frac{1}{2}FM\)? Wait, no, \(FM = 2\times MJ\), so \(MJ=\frac{FM}{2}=\frac{6}{2}=3\)? Wait, no, wait, let's re - check. The median is \(FJ\), so \(F\) is vertex, \(J\) is midpoint of \(DE\), and \(M\) is centroid on \(FJ\). So \(FM:MJ = 2:1\), so \(FJ=FM + MJ=2MJ+MJ = 3MJ\), and \(FM = 2MJ\). So if \(FM = 6\), then \(MJ=\frac{FM}{2}=\frac{6}{2}=3\)? Wait, no, that would mean \(FJ=6 + 3=9\), and \(FM:MJ = 6:3 = 2:1\), which is correct.

Step4: Find \(DK\)

Given \(DM = 12\), and \(DK\) is a median (since \(M\) is centroid, \(DK\) is a median). So, \(DM:MK = 2:1\), and \(DK=DM + MK\). Since \(DM = 2\times MK\), \(MK=\frac{DM}{2}=\frac{12}{2}=6\). Then \(DK=DM + MK=12 + 6 = 18\). Alternatively, since centroid divides the median into \(2:1\), the length from vertex to centroid is \(\frac{2}{3}\) of the median length. So \(DM=\frac{2}{3}DK\), so \(DK=\frac{3}{2}DM=\frac{3}{2}\times12 = 18\).

Answer:

\(DK = 18\), \(MJ = 3\), \(ML = 8\)