Question

“If a line parallel to a side of a triangle intersects the remaining sides in

two distince points, then the line divides the sides in the same proportion.”​

Answer 1

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.

theorem on similarity of triangles

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=AEEC

Proof:ar AEM=12×(AD)×(EM)

Similarly;ar BDE=12×(DB)×(EM)ar ADE=12×(AE)×(DN)ar DEC=12×(EC)×(DN)

Hence;

ar ADEar BDE=12×(AD)×(EM)12×(DM)=ADDB

Similarly;

ar ADE

ar 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=AEEC

proved