Question

1. in $delta pqr$, $sp = 42$ and $um = 17$. find $sm$, $pm$, $mr$, and $ur$.

$sm = \underline{\quad\quad}$
$pm = \underline{\quad\quad}$
$mr = \underline{\quad\quad}$
$ur = \underline{\quad\quad}$

Explanation:

Step1: Recall centroid ratio rule

In a triangle, the centroid (point $M$) divides each median into a ratio of $2:1$, where the segment from the vertex to the centroid is twice the segment from the centroid to the midpoint.

Step2: Calculate $SM$

$SP$ is a median, $M$ splits it into $PM:SM=2:1$. So $SM=\frac{1}{3}SP$
$SM=\frac{1}{3} \times 42 = 14$

Step3: Calculate $PM$

$PM=\frac{2}{3}SP$
$PM=\frac{2}{3} \times 42 = 28$

Step4: Calculate $UR$

$PU$ is a median, $M$ splits it into $UM:MU=1:2$, so $UR=UM + MU$, and $MU=2 \times UM$
$UR=17 + 2\times17 = 51$

Step5: Calculate $MR$

$PT$ is a median, $M$ splits it into $PM:MR=2:1$, so $MR=PM=28$ (correction: wait, no—actually, for median $PR$? No, $PT$ is median, $M$ is centroid, so $MR=2\times TM$, but since $PM=28$, and centroid divides $PR$? No, wait, $SP$ is median, $PM$ is part of median $PR$? No, correct: centroid divides each median into 2:1, so for median from $P$ to $QR$ (which is $PS$), we have $PM:MS=2:1$. For median from $R$ to $PQ$ (which is $RU$), $RM:MT=2:1$, but $MR$ is equal to $PM$? No, no—wait, $PM$ is 28, and since centroid divides the median into 2:1, $MR$ is equal to $PM$? No, no, $PM$ is part of median $PS$, $MR$ is part of median $RT$. Wait, no, the centroid divides all medians into 2:1, so for median $PR$? No, $T$ is midpoint of $PR$, so $PT=TR$, and $PM:MR=2:1$? No, $M$ is centroid, so $TM:MR=1:2$, but $PT=TR$, so $PR=2PT$, and $MR=\frac{2}{3}TR$? No, wait, no—actually, since $PM=28$, and in the centroid, the length from vertex to centroid is twice centroid to midpoint, so $MR=PM=28$? No, no, wait, $SP=42$, $PM=28$, $SM=14$. For $UM=17$, $MU=2\times17=34$, so $UR=UM+MU=17+34=51$. And $MR$: since $M$ is centroid, $MR$ is equal to $PM$? No, no, $MR$ is the segment from centroid to midpoint of $PQ$? No, $R$ to $M$ is the vertex to centroid, so $MR=2\times$ the segment from $M$ to midpoint of $PQ$. Wait, no, the given $UM=17$, which is the segment from centroid to midpoint of $PQ$, so $RU$ is the median, so $MR=2\times UM=2\times17=34$? Wait, no, $UM$ is from $U$ (midpoint) to $M$ (centroid), so $RU$ is the median, so $RM:MU=2:1$, so $RM=2\times UM=34$, and $UR=RM+MU=34+17=51$. And $MR=34$.

Correcting Step5:

Step5: Calculate $MR$

$RU$ is a median, $M$ splits it into $MR:UM=2:1$, so $MR=2\times UM$
$MR=2\times17=34$

Step6: Calculate $UR$

$UR=MR+UM$
$UR=34+17=51$

Answer:

$SM=14$
$PM=28$
$MR=34$
$UR=51$