In triangle PQR, PQ = 13, QR = 14, and PR = 18. Let M be the midpoint line QR. Find PM.
+128408 Let P = (0,0) Q = (13,0)
We can determine the coordiantes of R by finding the intersection of two circles
x^2 + y^2 = 18^2 → x^2 + y^2 = 324 (1)
(x -13)^2 + y^2 = 14^2 → x^2 - 26x + 169 + y^2 = 196 → x^2 - 26x + y^2 = 27 (2)
Subtract (2) from (1)
26x = 297
x = 297/26
And (297/26)^2 + y^2 = 324
y^2 = 324 - ( 297/26)^2
y^2 = 130815 / 676 take the positive root for y
y = sqrt (130815) / 26
R = ( 297/26 , sqrt (130815) /26)
M = ( (297 / 26 + 13) / 2 , sqrt (130815 / 52 ) = (635/52 , sqrt (130815) / 52 )
PM = (1/52) sqrt ( 635^2 + 130815) = (1/52) sqrt (534050) ≈ 14.05
