+1678 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 $PF$?
+129771 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)
