Question

Prove that if draw a line which is parallel to any side of a uiangle and intersect the other two sides at different points, then this line divides these two line in the same ratio.​

Answer 1

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Let a ∆ABC in which a line DE parallel to BC intersects AB at D and AC at E.

To prove DE divides the two sides in the same ratio.

AD/DB = AE/EC Construction ,

Join BE, CD Draw EF ⊥ AB and DG ⊥ AC.

We know that,

Area of triangle = ½ × base × height

Then,

ar( AADE) x AD~« EF

ar{ \BDE )

| frat

x DBx« EF

ar(AADE) AD ar( ABDE) DB

ar(AADE) AE

ai ADEC) EC ...{11)

Since, ∆BDE and ∆DEC lie between the same parallel DE and BC and are on the same base DE.

We have, area (∆BDE) = area(∆DEC) …..(iii)

From Equation (i), (ii) and (iii),

We get,

AD/DB = AE/EC

Hence proved.

Answer 2

Answer:

Let a ∆ABC in which a line DE parallel to BC intersects AB at D and AC at E.

To prove DE divides the two sides in the same ratio. AD/DB = AE/EC Construction: Join BE, CD

Draw EF ⊥ AB and DG ⊥ AC.

We know that, Area of triangle = ½ × base × height

Then, Since, ∆BDE and ∆DEC lie between the same parallel DE and BC and are on the same base DE.

We have, area (∆BDE) = area(∆DEC) …..(iii)

From Equation (i), (ii) and (iii),

We get, AD/DB = AE/EC Hence proved.

Step-by-step explanation: