Question

Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Hence in ΔPQR, prove that a line ℓ intersects the sides PQ and PR of a ∆PQR at L and M, respectively, such that LM || QR. If PL = 5.7 cm, PQ = 15.2 cm and MR = 5.5 cm, then find the length of PM (in cm).

Answer 1

Given:

ΔABC where DE || BC

To prove:

`(AD)/(DB) = (AE)/(EC)`

Construction:

Join BE and CD.

Draw DM ⊥ AC and EN ⊥ AB.

Proof:

`(ar(ADE))/(ar(BDE)) = (1/2 xx AD xx EN)/(1/2 xx DB xx EN)`

`(ar(ADE))/(ar(BDE)) = (AD)/(DB)`   ...(i)

`(ar(ADE))/(ar(DEC)) = (1/2 xx AE xx DM)/(1/2 xx EC xx DM)`

`(ar(ADE))/(ar(DEC)) = (AE)/(EC)`   ...(ii)

Now, ΔBDE and ΔDEC are on the same base DE and between the same parallel lines BC and DE.

∴ ar(BDE) = ar(DEC)

Hence, `(ar(ADE))/(ar(BDE)) = (ar(ADE))/(ar(DEC))`

`(AD)/(DB) = (AE)/(EC)`   ...[From equation (i) and (ii)]

Hence proved.

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

`(PL)/(LQ) = (PM)/(MR)`

`(PL)/(PQ - PL) = (PM)/(MR)`

`5.7/(15.2 - 5.7) = (PM)/5.5`

`5.7/9.5 = (PM)/5.5`

PM × 9.5 = 5.7 × 5.5

PM = `(5.7 xx 5.5)/9.5`

= `62.7/19`

= 3.3 cm