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 the area of ΔOMN : Area of ΔORQ.

Answer 1

(1)

Since `(PM)/(MQ) = 2/3`,

Let PM = 2x, MQ = 3x

Then,

PQ = PM + MQ

PQ = 2x + 3x

∴ PQ = 5x

Now in △PQR, MN || QR,

So, by the similarity of △PMN and △PQR,

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

`(MN)/(QR) = (2x)/(5x)`

∴ `(MN)/(QR) = 2/5`

(2)

Since MN || QR, and O lies on MR and QN,

• ∠OMN = ∠ORQ (because OM is along OR and MN || RQ)
• ∠ONM = ∠OQR (because ON is along OQ and MN || QR)

Thus, two angles of △OMN are respectively equal to two angles of △ORQ.

Therefore, by AA similarity,

△OMN ∼ △ORQ

Hence proved.

(3)

From part (2),

△OMN ∼ △ORQ

Hence, the ratio of their areas is the square of the ratio of their corresponding sides:

`"area(ΔOMN)"/"area(ΔORQ​)" = ((MN)/(RQ))^2`

From part (1),

`(MN)/(RQ) = 2/5`

So,

`"area(ΔOMN)"/"area(ΔORQ​)" = (2/5)^2`

`"area(ΔOMN)"/"area(ΔORQ​)" = 4/25`

∴ area(ΔOMN) : area(ΔORQ​) = 4 : 25

Hence,

(1) `(MN)/(QR) = 2/5`.

(2) ΔOMN and ΔORQ are similar.

(3) The area of ΔOMN : Area of ΔORQ = 4 : 25.