Question

geometry
worksheet 4.6b – medians & centroids
page 1
median – a segment connecting the vertex of a triangle to the midpoint of the opposite side.
centroid – point where all medians of a triangle intersect. it is also the “balancing point” for the triangle. each median is cut into two segments with a ratio of 2:1 (the longer segment is between the vertex & the centroid).

1. in $\triangle xyz$, $\overline{yw}$ is a median. what is $xw$ if $xz = 17$?
2. in $\triangle abc$, $\overline{bx}$, $\overline{cz}$, and $\overline{ay}$ are medians. if $ax = 3x - 9$, $xc = 2x - 4$, and $zb = 2x + 1$, what is $az$?

in $\triangle def$, $\overline{ds}$, $\overline{fr}$, and $\overline{et}$ are medians.

1. find $ev$ if $vt = 5$.
2. if $fr = 20.1$, what is the measure of $\overline{vr}$?

in $\triangle tuv$, $\overline{te}$, $\overline{ud}$, and $\overline{vc}$ are medians.

1. find $ev$ if $uv = 24$.
2. if $tc = 8$, find $tu$.
3. what is $td$ if $tv = 29$?

in $\triangle mnp$, $my$, $px$, and $nz$ are medians.

1. find the measure of $\overline{wy}$ if $mw = 22$.
2. what is $nw$ if $zw = 10$?
3. if $pw = 13$, what is $wx$?

in $\triangle fgh$, $\overline{fj}$, $\overline{hi}$, and $\overline{gk}$ are medians.

1. what is $xk$ if $gk = 13.5$?
2. if $fx = 10.6$, what is the measure of $\overline{xj}$?
3. find $hx$ if $hi = 9$.

Explanation:

Step1: Use median midpoint property

$XW = \frac{1}{2}XZ$
$XW = \frac{1}{2} \times 17 = 8.5$

Step2: Set $AX=XC$ to solve $x$

$3x-9=2x-4$
$3x-2x=-4+9 \implies x=5$
Find $ZB$: $ZB=2(5)+1=11$
Use centroid ratio $AZ:ZB=2:1$
$AZ=2 \times ZB = 2 \times 11=22$

Step3: Use centroid ratio $EV:VT=2:1$

$EV=2 \times VT$
$EV=2 \times 5=10$

Step4: Use centroid ratio $VR:FR=1:3$

$VR=\frac{1}{3}FR$
$VR=\frac{1}{3} \times 20.1=6.7$

Step5: Use centroid ratio $UV:EV=2:1$

$EV=\frac{1}{2}UV$
$EV=\frac{1}{2} \times 24=12$

Step6: Use centroid ratio $TC:TU=1:3$

$TU=3 \times TC$
$TU=3 \times 8=24$

Step7: Use centroid ratio $TD:TV=2:3$

$TD=\frac{2}{3}TV$
$TD=\frac{2}{3} \times 29=\frac{58}{3} \approx 19.33$

Step8: Use centroid ratio $MW:WY=2:1$

$WY=\frac{1}{2}MW$
$WY=\frac{1}{2} \times 22=11$

Step9: Use centroid ratio $ZW:NW=1:2$

$NW=2 \times ZW$
$NW=2 \times 10=20$

Step10: Use centroid ratio $PW:WX=2:1$

$WX=\frac{1}{2}PW$
$WX=\frac{1}{2} \times 13=6.5$

Step11: Use centroid ratio $XK:GK=1:3$

$XK=\frac{1}{3}GK$
$XK=\frac{1}{3} \times 13.5=4.5$

Step12: Use centroid ratio $FX:XJ=2:1$

$XJ=\frac{1}{2}FX$
$XJ=\frac{1}{2} \times 10.6=5.3$

Step13: Use centroid ratio $HX:HI=2:3$

$HX=\frac{2}{3}HI$
$HX=\frac{2}{3} \times 9=6$

Answer:

1. $8.5$
2. $22$
3. $10$
4. $6.7$
5. $12$
6. $24$
7. $\frac{58}{3}$
8. $11$
9. $20$
10. $6.5$
11. $4.5$
12. $5.3$
13. $6$