In ΔABC, D and E are the points on sides AB and AC respectively.

Find whether DE || BC, if

(i) AB = 9 cm, AD = 4 cm, AE = 6 cm and EC = 7.5 cm.

(ii) AB = 6.3 cm, EC = 11.0 cm, AD = 0.8 cm and EA = 1.6 cm.

(i) From figure,

BD = AB - AD = 9 - 4 = 5 cm

In ∆ADE and ∆ABC,

$\dfrac{AE}{EC} = \dfrac{6}{7.5} = \dfrac{4}{5} \\[1em] \dfrac{AD}{BD} = \dfrac{4}{5} \\[1em] \text{Since } \dfrac{AE}{EC} = \dfrac{AD}{BD}.$

Hence, DE || BC by the converse of Basic proportionality theorem.

(ii) From figure,

BD = AB - AD = 6.3 - 0.8 = 5.5 cm

In ∆ADE and ∆ABC,

$\dfrac{AE}{EC} = \dfrac{1.6}{11} = \dfrac{0.8}{5.5} \\[1em] \dfrac{AD}{BD} = \dfrac{0.8}{5.5} \\[1em] \text{Since } \dfrac{AE}{EC} = \dfrac{AD}{BD}.$

Hence, DE || BC by the converse of Basic proportionality theorem.