Question

MP is a median of the triangle MNO. NQ bisects MP. NQ meets MO in R. MO = 30cm. Find the length of MR.

a) 8cm. b) 10cm. c) 12cm. d) 15cm.

Answer 1

There are two methods given here. One is by areas and another by vectors. Vector method is simple.

MR : RO = 1 : 2.
So MR = 30/3= 10cm.

Perhaps more methods exist ..

Answer 2

Let M is zero vector .
Q is middle in NO
so, Q is vector ( M + O)/2
E is the middle point of MQ.
so, E is vector {M+ O }/4
eqn of MO :
r = (1 - s) 0 + sO _________(1)
where s is constant.
r = sO
again, eqn of QE
r = (1 -t) Q +t (M+ O)/4_____(2)
where t is constant
solve eqns (1) and (2)
t = 4/3 and s = 1/3
hence, eqn of MO :
r = 1/3 O________(3)

now, eqn of MO from section formula ,
r = ß O /(ß + 1)______(4)

compare (3) and (4)
ß /(ß + 1) = 1/3
ß = 1/2
hence,
MR/MO = 1/3
MR/30 = 1/3
MR = 10 cm