Math / Geometry

QuestionLas medianas del $\triangle D E F$ son $\overline{D K}, \overline{E L}$ y $\overline{F J}$ . Se encuentran en un punto único $M$ . (En otras palabras, $M$ es el baricentro del $\triangle D E F$ ). Suponer que $M L=9, M J=8$ y $D K=24$ . Hallar las longitudes siguientes. Observar que la figura no está trazada a escala. $\begin{array}{c} F J= \\ D M= \\ E M= \end{array}$

Studdy Solution

STEP 1

1. $M$ is the centroid (barycenter) of $\triangle DEF$ .
2. The centroid divides each median into a ratio of $2:1$ , with the longer segment being between the vertex and the centroid.
3. Given $ML = 9$ , $MJ = 8$ , and $DK = 24$ .

STEP 2

1. Use the properties of the centroid to find $FJ$ .
2. Use the properties of the centroid to find $DM$ .
3. Use the properties of the centroid to find $EM$ .

STEP 3

The centroid $M$ divides each median into two segments, with the ratio $2:1$ . This means the segment from the vertex to the centroid is twice as long as the segment from the centroid to the midpoint of the opposite side.
For $\overline{FJ}$ , given $MJ = 8$ , the entire median $FJ$ is:
$FJ = 3 \times MJ$ $FJ = 3 \times 8 = 24$

STEP 4

For $\overline{DK}$ , given $DK = 24$ , we know $DM$ is twice the length of $MK$ . Since $DK$ is the entire median:
$DM = \frac{2}{3} \times DK$ $DM = \frac{2}{3} \times 24 = 16$

STEP 5

For $\overline{EL}$ , given $ML = 9$ , the entire median $EL$ is:
$EL = 3 \times ML$ $EL = 3 \times 9 = 27$
Since $EM$ is twice $ML$ :
$EM = 2 \times ML$ $EM = 2 \times 9 = 18$
The lengths are: $FJ = 24$ $DM = 16$ $EM = 18$

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