Question

In the given figure pqrs is a parallelogram diagonals PR and qs intersect at o . l is a on PR LP/ po= 2/3 .ql meets sp produced at m . Find (1) lm / mq (2) ar (∆lmp)/ar(∆lgr)

Answer 1

Answer:

Lets write the known facts first,

PQRS is a parallelogram,

PL/PO = 2/3

Now, In Triangle PLM and Triangle QLR

Angle QLR = Angle PLM (Vertically opposite angles)

Angle LPM = Angle LRQ ( Alternate Interior Angles)

Angle LMP = Angle LQR ( Alternate Interior Angles)

So, Triangle LPM is Similar to Triangle LQR

Let LP = 2x,

So PO = 3x, OL = X

PO = RO (Diagonals in a parallelogram bisect each other)

Using above Data

LP = 2x

LR = x + 3x = 4x

So, The ratio of Sides of the Triangles comes out to be 1:2.

1) Therefore, LM/LQ = 1/2

So, LM/MQ = 1/3

2) As ratio of Sides of Triangles = 1:2

Ratio of their Areas = $(1:2)^2$ = (1:4)