Question

1. if g is the centroid of △ace, ag = 8, gf = 7, and bg = 5, find each measure.

a) gd =

b) ad =

c) cg =

d) ge =

e) be =

1. if x is the centroid of △rst, tu = 27, sw = 18, and rv = 21, find each measure.

a) tx =

b) xu =

c) sx =

d) xw =

e) rx =

Explanation:

Problem 3 (ΔACE, centroid G):

Recall: The centroid divides each median into a 2:1 ratio, with the longer segment between the vertex and centroid.

Step1: Find GD (median AD)

AG = 8 is 2 parts, so 1 part = $\frac{8}{2}=4$. GD is 1 part.
$GD = 4$

Step2: Find AD (total length)

AD = AG + GD
$AD = 8 + 4 = 12$

Step3: Find CG (median CD)

BG = 5 is 1 part, so CG is 2 parts.
$CG = 2\times5 = 10$

Step4: Find GE (median BE)

GF = 7 is 1 part, so GE is 2 parts.
$GE = 2\times7 = 14$

Step5: Find BE (total length)

BE = BG + GE
$BE = 5 + 14 = 19$

Problem 4 (ΔRST, centroid X):

Recall: The centroid divides each median into a 2:1 ratio, with the longer segment between the vertex and centroid.

Step6: Find TX (median RV)

RV = 21, total 3 parts. 1 part = $\frac{21}{3}=7$. TX is 2 parts.
$TX = 2\times7 = 14$

Step7: Find XU (median TU)

TU = 27, total 3 parts. 1 part = $\frac{27}{3}=9$. XU is 1 part.
$XU = 9$

Step8: Find SX (median SW)

SW = 18, total 3 parts. 1 part = $\frac{18}{3}=6$. SX is 2 parts.
$SX = 2\times6 = 12$

Step9: Find XW (median SW)

XW is 1 part of SW.
$XW = 6$

Step10: Find RX (median RV)

RX is 1 part of RV.
$RX = 7$

Answer:

Problem 3:

a) $GD = 4$
b) $AD = 12$
c) $CG = 10$
d) $GE = 14$
e) $BE = 19$

Problem 4:

a) $TX = 14$
b) $XU = 9$
c) $SX = 12$
d) $XW = 6$
e) $RX = 7$