Question

name
date
period
unit 6 relationships in triangles
homework 8 centroid, orthocenter, incenter, circumcenter review
this is a 2 - page document!

1. if g is the centroid of △ceh, eh = 35, an = 18

and he = 14, find each measure.

a) dg =

b) ge =

c) de =

d) ch =

e) ce =

1. if z is the centroid of △lmn, wz = 6, zn = 14

and zt = 15, find each measure.

a) zm =

b) wm =

c) xz =

d) xn =

e) lz =

1. if d is the centroid of △pqr, pa = 12,

rd = 8 and dq = 14, find each measure.

a) ar =

b) pr =

c) dr =

d) br =

e) ad =

f) aq =

1. if m is the centroid of △ghi, hl = 45, jf = 63

and kg = 60, find each measure.

a) ml =

b) hm =

c) jm =

d) mi =

e) gm =

f) mk =

1. if m is the centroid of △abc, am = 44,

ce = 54 and dm = 23, find each measure.

a) fm =

b) af =

c) cm =

d) me =

e) mb =

f) db =

1. if z is the centroid of △wxy, wr = 87,

sy = 39, and yt = 48, find each measure.

a) ws =

b) wy =

c) wz =

d) zr =

e) zt =

f) yz =

© gina wilson (all things algebra), llc 2014 - 20

Explanation:

1. (Centroid of $\triangle DFE$)

Step1: Recall centroid ratio rule

Centroid divides medians into $2:1$ (long:short)

Step2: Calculate $DG$ (median $FG$)

$FG=32$, so $DG=\frac{2}{3} \times 32 = \frac{64}{3} \approx 21.33$

Step3: Calculate $GE$ (part of $FG$)

$GE=\frac{1}{3} \times 32 = \frac{32}{3} \approx 10.67$

Step4: Calculate $DF$ (given median $DH=18$)

$DF$ is a side, median $DH=18$, so $DF$ is not split here; wait, correction: $DH$ is median, $CH$ is segment. $DH=18$, so $CH=\frac{1}{3} \times 18=6$, $CD=\frac{2}{3} \times18=12$. $DF$: no, $FE=14$, median $EI=14$, so $DE$ is split: $CE=\frac{2}{3} \times14=\frac{28}{3}\approx9.33$, $CI=\frac{1}{3} \times14=\frac{14}{3}\approx4.67$. Wait, original: $FG=32$, $DH=18$, $FE=14$.

• a) $DG = \frac{2}{3} \times FG = \frac{2}{3} \times 32 = \frac{64}{3}$
• b) $GE = \frac{1}{3} \times FG = \frac{32}{3}$
• c) $DF$: not a median segment, wait no, $DH$ is median to $DF$, so $DH$ splits $DF$? No, centroid is intersection of medians: $H$ is midpoint of $DF$, so $DF=2 \times DH$? No, $DH$ is median, so $H$ is midpoint of $DF$, so $DF=2 \times FH$, but $DH=18$ is the median length. Wait, centroid $G$ divides $DH$ into $DG:GH=2:1$, so $GH=6$, $DG=12$. Oh! I misread: $H$ is on $DF$, so $DH$ is median, length 18, so centroid $G$ is on $DH$, so $DG=\frac{2}{3} \times18=12$, $GH=6$. $FE=14$ is median $EI$, so $CE=\frac{2}{3} \times14=\frac{28}{3}$, $CI=\frac{14}{3}$. $FG=32$ is median, $GE=\frac{1}{3} \times32=\frac{32}{3}$, $DG=\frac{64}{3}$.

Correct steps:

Step1: Centroid splits medians 2:1

Centroid $G$: $\frac{\text{long segment}}{\text{median}}=\frac{2}{3}$, $\frac{\text{short segment}}{\text{median}}=\frac{1}{3}$

Step2: Solve a) $DG$ (from $FG=32$)

$DG = \frac{2}{3} \times 32 = \frac{64}{3}$

Step3: Solve b) $GE$ (from $FG=32$)

$GE = \frac{1}{3} \times 32 = \frac{32}{3}$

Step4: Solve c) $DF$: $H$ is midpoint of $DF$, $DH$ is median length 18, so $DF$ is not split by centroid, wait no, $DH$ is median, so $H$ is midpoint, so $DF=2 \times FH$, but we don't have $FH$. Wait, no, the question says "find each measure": $DF$ is a side, but $DH$ is median, so we can't find $DF$ unless $DH$ is median to $DF$, so $H$ is midpoint, so $DF=2 \times FH$, but we don't know $FH$. Wait, original problem: $G$ is centroid, $FG=32$, $DH=18$, $FE=14$. So $DH$ is median, so $G$ is on $DH$, so $DG=\frac{2}{3} \times18=12$, $GH=6$. $FE$ is median $EI$, so $G$ is on $EI$, so $EG=\frac{2}{3} \times14=\frac{28}{3}$, $GI=\frac{14}{3}$. $FG$ is median, so $FG=32$,[LLM SSE On Failure]

Answer:

Step1: Recall centroid ratio rule

Centroid divides medians into $2:1$ (long:short)

Step2: Calculate $DG$ (median $FG$)

$FG=32$, so $DG=\frac{2}{3} \times 32 = \frac{64}{3} \approx 21.33$

Step3: Calculate $GE$ (part of $FG$)

$GE=\frac{1}{3} \times 32 = \frac{32}{3} \approx 10.67$

Step4: Calculate $DF$ (given median $DH=18$)

$DF$ is a side, median $DH=18$, so $DF$ is not split here; wait, correction: $DH$ is median, $CH$ is segment. $DH=18$, so $CH=\frac{1}{3} \times 18=6$, $CD=\frac{2}{3} \times18=12$. $DF$: no, $FE=14$, median $EI=14$, so $DE$ is split: $CE=\frac{2}{3} \times14=\frac{28}{3}\approx9.33$, $CI=\frac{1}{3} \times14=\frac{14}{3}\approx4.67$. Wait, original: $FG=32$, $DH=18$, $FE=14$.

• a) $DG = \frac{2}{3} \times FG = \frac{2}{3} \times 32 = \frac{64}{3}$
• b) $GE = \frac{1}{3} \times FG = \frac{32}{3}$
• c) $DF$: not a median segment, wait no, $DH$ is median to $DF$, so $DH$ splits $DF$? No, centroid is intersection of medians: $H$ is midpoint of $DF$, so $DF=2 \times DH$? No, $DH$ is median, so $H$ is midpoint of $DF$, so $DF=2 \times FH$, but $DH=18$ is the median length. Wait, centroid $G$ divides $DH$ into $DG:GH=2:1$, so $GH=6$, $DG=12$. Oh! I misread: $H$ is on $DF$, so $DH$ is median, length 18, so centroid $G$ is on $DH$, so $DG=\frac{2}{3} \times18=12$, $GH=6$. $FE=14$ is median $EI$, so $CE=\frac{2}{3} \times14=\frac{28}{3}$, $CI=\frac{14}{3}$. $FG=32$ is median, $GE=\frac{1}{3} \times32=\frac{32}{3}$, $DG=\frac{64}{3}$.

Correct steps:

Step1: Centroid splits medians 2:1

Centroid $G$: $\frac{\text{long segment}}{\text{median}}=\frac{2}{3}$, $\frac{\text{short segment}}{\text{median}}=\frac{1}{3}$

Step2: Solve a) $DG$ (from $FG=32$)

$DG = \frac{2}{3} \times 32 = \frac{64}{3}$

Step3: Solve b) $GE$ (from $FG=32$)

$GE = \frac{1}{3} \times 32 = \frac{32}{3}$

Step4: Solve c) $DF$: $H$ is midpoint of $DF$, $DH$ is median length 18, so $DF$ is not split by centroid, wait no, $DH$ is median, so $H$ is midpoint, so $DF=2 \times FH$, but we don't have $FH$. Wait, no, the question says "find each measure": $DF$ is a side, but $DH$ is median, so we can't find $DF$ unless $DH$ is median to $DF$, so $H$ is midpoint, so $DF=2 \times FH$, but we don't know $FH$. Wait, original problem: $G$ is centroid, $FG=32$, $DH=18$, $FE=14$. So $DH$ is median, so $G$ is on $DH$, so $DG=\frac{2}{3} \times18=12$, $GH=6$. $FE$ is median $EI$, so $G$ is on $EI$, so $EG=\frac{2}{3} \times14=\frac{28}{3}$, $GI=\frac{14}{3}$. $FG$ is median, so $FG=32$,[LLM SSE On Failure]