Quick Question

Points M, N, and O are the midpoints of sides KL, LJ, and JK, respectively, of triangle JKL. Points P, Q, and R are the midpoints of NO, OM, and MN, respectively. If the area of triangle PQR is 27, then what is the area of triangle LPQ?

Note:

The line joining the midpoints of a two sides of a triangle is parallel to the 3rd side. You can look up the proof if you wish.

So

KJ is parallel to MN is parallel to QP

ans

KL is parallel to ON is parallel to QR

So every triangle in this figure is similar to all the others.

LPQ=27u^2

So the area of MNO = 4*27 =  108u^2

And the area of LJK = 4*108 = 432u^2

Before I think on it any further can you confirm that it really is triangle LPQ that you want the area of ...

Here is the diagram.

ok

triangle LMN is congruent to triangle ONM  so LM=NO

And NP is a half of NO

And RQ=NP

So the horizontal distance from Q to L is 3 times the horizontal distance from Q to R

Triangles  RPQ and LPQ share the same base, that is PQ

But the perpendicular height of LPQ is three times that of RPQ 

Since the area of triangle RPQ is  27,

It follows that the area of LPQ is  3*27=  81u^2