The vertices of triangle PQR are # P(-4, -1), Q(2,9), and R(6,3)#. #S# is the midpoint of #\bar(PQ)# and #T# is the midpoint of #\bar (QR)#. How do you prove that #\bar(ST)# is || #\bar (PR)# and #ST = 1/2 PR#?

1 Answer
Dec 6, 2017

Please see below.

Explanation:

We have #P(-4,-1)# and #R(6,3)# and therefore slope of #PR# is #(6-(-4))/(3-(-1))=10/4=5/2#.

Further, length of #bar(PR)=sqrt((6-(-4))^2+(3-(-1))^2)# or #sqrt(10^2+4^2)=sqrt116=2sqrt29#

As #S# is the midpoint of line joining #P(-4,-1)# and #Q(2,9)# i.e. its coordinates are #((-4+2)/2,(-1+9)/2)# i.e. we have #S(-1,4)#.

Similarly as #T# is the midpoint of line joining #Q(2,9)# and #R(6,3)# i.e. its coordinates are #((2+6)/2,(9+3)/2)# i.e. we have #T(4,6)#.

As slope of #bar(ST)# joining #S(-1,4)# and #T(4,6)# is #(6-4)/(4-(-1))=2/5#, same as that of #PR#

hence #bar(ST)#||#bar(PR)#.

Further, #ST=sqrt((4-(-1))^2+(6-4)^2)-sqrt(5^2+2^2)=sqrt29# and as #PR=2sqrt29#,

#ST=1/2PR#