Question

if a line is drawn parallel to one side of a triangle ,to intersect the other two sides ,the other two sides are divided in the same ratio using above theorem solve the following. ....1. .. in a triangle abc ,D and E are the points in Ab and ac such that De//bc ad=2, ab =6cm,ac=9cm find ae

Answer 1

Construction: ABC is a triangle. DE || BC and DE intersects AB at D and AC at E.

Join B to E and C to D. Draw DN ⊥ AB and EM ⊥ AC.

To prove: ADDB=AEECADDB=AEEC

Proof:

ar AEM=12×(AD)×(EM)ar AEM=12×(AD)×(EM)

Similarly;

ar BDE=12×(DB)×(EM)ar BDE=12×(DB)×(EM)

ar ADE=12×(AE)×(DN)ar ADE=12×(AE)×(DN)

ar DEC=12×(EC)×(DN)ar DEC=12×(EC)×(DN)

Hence;

ar ADEar BDE=12×(AD)×(EM)12×(DB)×(EM)=ADDBar ADEar BDE=12×(AD)×(EM)12×(DB)×(EM)=ADDB

Similarly;

ar ADEar DEC=AEECar ADEar DEC=AEEC

Triangles BDE and DEC are on the same base, i.e. DE and between same parallels, i.e. DE and BC.

Hence, ar(BDE) = ar(DEC)

From above equations, it is clear that;

ADDB=AEECADDB=AEEC proved