D and E are points on the sides AB and AC respectively of ΔABC. For each of the following cases, state whether DE ∥ BC :

(i) AD = 5.7 cm, BD = 9.5 cm, AE = 3.6 cm and EC = 6 cm.

(ii) AB = 5.6 cm, AD = 1.4 cm, AC = 9.6 cm and EC = 2.4 cm.

(iii) AB = 11.7 cm, BD = 5.2 cm, AE = 4.4 cm and AC = 9.9 cm.

(iv) AB = 10.8 cm, BD = 4.5 cm, AC = 4.8 cm and AE = 2.8 cm.

By basic proportionality theorem,

A line drawn parallel to a side of triangle divides the other two sides proportionally.

(i) Given,

AD = 5.7 cm

BD = 9.5 cm

AE = 3.6 cm

EC = 6 cm.

Check for proportionality,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{5.7}{9.5} \\[1em] \Rightarrow \dfrac{AD}{DB} = \dfrac{3}{5} = 0.6 \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{3.6}{6.0} \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{3}{5} = 0.6 \\[1em] \therefore \dfrac{AD}{DB} = \dfrac{AE}{EC}.$

We conclude that DE is parallel to BC

Hence, DE is parallel to BC.

(ii) Given,

AB = 5.6 cm

AD = 1.4 cm

AC = 9.6 cm

EC = 2.4 cm.

From figure,

DB = AB - AD = 5.6 - 1.4 = 4.2 cm

AE = AC - EC = 9.6 - 2.4 = 7.2 cm

Check for proportionality,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{1.4}{4.2} \\[1em] \Rightarrow \dfrac{AD}{DB} = \dfrac{1}{3} \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{7.2}{2.4} \\[1em] \Rightarrow \dfrac{AE}{EC} = 3 \\[1em] \therefore \dfrac{AD}{DB} \ne \dfrac{AE}{EC}.$

We conclude that DE is not parallel to BC

Hence, DE is not parallel to BC.

(iii) Given,

AB = 11.7 cm

BD = 5.2 cm

AE = 4.4 cm

AC = 9.9 cm.

From figure,

AD = AB - BD = 11.7 - 5.2 = 6.5 cm

EC = AC - AE = 9.9 - 4.4 = 5.5 cm

Check for proportionality,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{6.5}{5.2} \\[1em] \Rightarrow \dfrac{AD}{DB} = \dfrac{5}{4} = 1.25 \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{4.4}{5.5} \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{4}{5} = 0.8 \\[1em] \therefore \dfrac{AD}{DB} \ne \dfrac{AE}{EC}.$

We conclude that DE is not parallel to BC

Hence, DE is not parallel to BC.

(iv) Given,

AB = 10.8 cm

BD = 4.5 cm

AC = 4.8 cm

AE = 2.8 cm.

AD = AB - BD = 10.8 - 4.5 = 6.3 cm

EC = AC - AE = 4.8 - 2.8 = 2 cm

Check for proportionality,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{6.3}{4.5} \\[1em] \Rightarrow \dfrac{AD}{DB} = \dfrac{7}{5} = 1.4 \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{2.8}{2.0} \\[1em] \Rightarrow \dfrac{AE}{EC} = \dfrac{7}{5} = 1.4 \\[1em] \therefore \dfrac{AD}{DB} = \dfrac{AE}{EC}.$

We conclude that DE is parallel to BC

Hence, DE is parallel to BC.