Question

In ΔPQR, MN is parallel to QR and `(PM)/(MQ) = 2/3`

1) Find `(MN)/(QR)`

2) Prove that ΔOMN and ΔORQ are similar.

3) Find, Area of ΔOMN : Area of ΔORQ

Answer 1

In ΔPMN and ΔPQR, MN || QR

`=> angle PMN =anglePQR`   (alternate angles)

`=> angle PNM = anglePRQ`  (alternate angles)

=> ΔPMN ~ ΔPQR (AA postulate)

`= (PM)/(PQ) = (MN)/(QR)`

`=> 2/5 = (MN)/(QR)    [(PM)/(MQ) = 2/3 => (PM)/(PQ) = 2/5]`

2) In ΔOMN and ΔORQ,

`angleOMN = angleORQ`   (alternate angles)

`angleMNO = angleOQR`    (alternate angles)

=> ΔOMN ~ ΔORQ  (AA postulate)

3) `"Area of ΔOMN"/"Area of ΔORQ" = (MN)/(RQ) = 2/5`