Question

1. in $\triangle xyz$ shown below, medians $xe$, $xf$, and $zd$ intersect at $c$.

if $ce = 4$, $yf = 18$, and $xz = 12$, determine and state the perimeter of triangle $cfx$.

1. given $\triangle mro$ shown below, with trapezoid $ptro$,

$mr = 9$, $mp = 2$, and $po = 4$.

what is the length of $overline{tr}$?
a) 4.5 b) 5 c) 3 d) 6

Explanation:

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Problem 17

Step1: Find length of XC

The centroid divides a median into a 2:1 ratio, so $XC = 2 \times CE$.
$XC = 2 \times 4 = 8$

Step2: Find length of CF

XF is a median, so $YF = XF = 18$. The centroid splits XF into 2:1, so $CF = \frac{1}{3} \times XF$.
$CF = \frac{1}{3} \times 18 = 6$

Step3: Identify length of XZ

Given $XZ = 12$, this is a side of $\triangle CFX$.

Step4: Calculate perimeter

Perimeter = $XC + CF + XZ$.
$\text{Perimeter} = 8 + 6 + 12 = 26$

Step1: Set up proportion

Since $PT \parallel OR$ (trapezoid property), $\triangle MPT \sim \triangle MOR$ by AA similarity. The ratio of corresponding sides equals the ratio of segment lengths on the median: $\frac{MP}{MR} = \frac{MT}{MO}$.
$\frac{2}{9} = \frac{MT}{MT + 4}$

Step2: Solve for MT

Cross-multiply to solve the equation.
$2(MT + 4) = 9MT$
$2MT + 8 = 9MT$
$8 = 7MT$
$MT = \frac{8}{7}$ (not needed directly; use ratio for TR)

Step3: Find length of TR

Use the similarity ratio: $\frac{MP}{MR} = \frac{MT}{MO} = \frac{PT}{OR}$, and $TR = MR - MT$. Alternatively, use segment ratio: $\frac{MP}{PR} = \frac{MT}{TR}$, where $PR = MR - MP = 9-2=7$.
$\frac{2}{7} = \frac{2}{TR}$
$TR = 7$ (correction: use $\frac{MP}{MR} = \frac{MT}{MO} \implies MO = MT + 4$, so $\frac{2}{9} = \frac{MT}{MT+4} \implies 2MT +8=9MT \implies 7MT=8 \implies MT=\frac{8}{7}$. Then $TR=MR - MT=9-\frac{8}{7}=\frac{55}{7}$ is wrong. Correct approach: $\frac{MP}{MO} = \frac{MT}{MR}$: $\frac{2}{2+4}=\frac{MT}{9} \implies \frac{2}{6}=\frac{MT}{9} \implies MT=3$. Then $TR=9-3=6$)

Answer:

26

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Problem 18