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In the diagram, $\triangle PQR$ is right-angled at $P$ and has $PQ=2$ and $PR=5$.  Altitude $PL$ intersects median $RM$ at $F$.  What is the length of $PF$?

 

 Jan 3, 2024
 #1
avatar+129193 
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Let R =(0,5)    ,  O = (2,0)    M = (1,0)

 

The line through RQ  has a slope of    (-5/2)

 

The line through PL is perpendicular to the first line so it has a slope of 2/5   and it  passes through (0,0) so its equation is   y= (2/5)x       (1)

 

The line through  RM  has a slope of  -5   and passes through  (0,5)

So its equation is   y =-5x + 5         (2)

 

Equatiing (1) and (2)  to find the x coordinate of  F

-5x + 5  =(2/5)x                muliply hrough by 5

 

-25x + 25 = 2x 

 

25 = 27x

 

x = 25 / 27        and  y =  (2/5)(25/27)  =  10/27

 

PF =  sqrt  [ (25/27)^2 + ( 10/27)^2 ]   =  sqrt [ 25^2 + 10^2] / 27  = sqrt [ 725 ] / 27 =  

 

sqrt [ 25 * 29 ] /27  =  (5/27)sqrt (29)

 

 

cool cool cool

 Jan 3, 2024

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