Question

In the given figure, PS = 3RS. M is the midpoint of QR. If TR || MN || QP, then prove that:

ST = `(1)/(3)"LS"`

Answer 1

Proof :
In ΔPQR,
Since M is the mid-point of QR, and MN || QP, N is the mid-point of PR.
⇒ PN = PR
Given PS = 3RS
⇒ PS = RS = PN + NR + RS
But, PS = PN + NR + Rs
⇒ PN = PR = Rs
⇒R is the mid-point of SN
RT || MN
⇒ T is the mid-point of SM  ....(i)
Also, N is the mid-point of PR and MN || LP
⇒ M is the mid-point of LT   ....(ii)
So, from (i) and (ii),
LM = MT = ST

⇒ ST = `(1)/(3)"LS"`.