AC is a diagonal too...& plz zoom into the picture ....

Geometry Level 2

In this Parallelogram $ABCD$ , $BD$ is a diagonal.

In Triangle $ABD$ , Point $P$ is the centroid.

In Triangle $BCD$ , Point $Q$ is the centroid.

Which line segment is equal to $AP$ ?

PO BP DE PQ

1 solution

Harshendu Mahto
Jan 22, 2016

In every triangle the centroid divides each median into two parts in the ratio of 2:1.

So, in triangle ABD, AP = 2/3 AO [i]

and PO = 1/3 AO [ii]

in triangle BCD, CQ = 2/3 CO [iii]

and QO = 1/3 CO [iv]

But , AO = CO [diagonals of a //gm bisect each other]

Therefore,

2/3 AO = 2/3 CO

=> AP = CQ {From [i] and [iii] }

Adding [ii] and [iv] ,

1/3 AO + 1/3 CO = PO + QO

=> 1/3 (AO + CO) = PQ

=> 1/3 AC = PQ

Now,

AC = AP + CQ + PQ

AC = 2AP + PQ [ AP = CQ ]

AC = 2AP + 1/3 AC [PQ = 1/3 AC]

AC - 1/3 AC = 2AP

2/3 AC = 2AP

Therefore,

AP = 1/3 AC

But, AP = CQ

So, CQ = 1/3 AC

and, PQ = 1/3 AC [proved above]

Therefore,

AP = PQ = CQ

Great solution bro. You are a genius

abhigyan adarsh - 5 years, 3 months ago