Question

The medians of $▵DEF$are shown. Find the lengths indicated.

1. $EP=?$
2. $PR=?$
3. If $FT=9,$ then $PT=?$and $FP=?$

Short answer

$EP=10,PR=4,FP=6.$

Step-by-step solution

Step 1. Given information:

Medians of triangle$DEF$are given:

$FT=9$

Step 2. Concept used:

We use basic geometric and angle concept.

Step 3. Applying the concept:

a.

From the figure, and the theorem $10.4,$

$\begin{array}{l}PS=\frac{1}{3}ES,\\ \Rightarrow ES=EP+PS.\\ \text{Now,if}EP=x,\\ ES=x+5,\\ \Rightarrow 5=\frac{1}{3}(x+5),\\ \Rightarrow 15=x+5\\ \Rightarrow x=10\\ \Rightarrow EP=10.\end{array}$

b.

From the figure, and the theorem $10.4,$

$\begin{array}{l}DP=\frac{2}{3}DR,\\ \Rightarrow DR=DP+PR.\\ \text{Now,if}PR=y,\\ DR=8+y,\\ \Rightarrow 8=\frac{2}{3}(8+y),\\ \Rightarrow 24=16+2y,\\ \Rightarrow 2y=8,\\ \Rightarrow y=4,\\ \Rightarrow PR=4.\end{array}$

c.

From the figure, and the theorem $10.4,$,

$\begin{array}{l}FP:PT:FT=2:1:3.\\ \text{Anditisalsogiventhat}FT=9,\\ \Rightarrow PT:9=1:3,\\ \Rightarrow 3PT=9,\\ \Rightarrow PT=\frac{9}{3},\\ \Rightarrow PT=3.\\ \text{Since}FP:9=2:3,\\ \Rightarrow FP=\frac{2}{3}\times 9,\\ \Rightarrow FP=\frac{18}{3},\\ \Rightarrow FP=6.\end{array}$