Question

k In the given figure, AD = 3 cm, AE = 5 cm,
BD = 4 cm, CE = 4 cm, CF = 2 cm, BF = 2.5
cm, then find the pair of parallel lines and
hence their lengths.
[4]​

Answer 1

$\\ \huge\bf\underline{ Required \: Question :- }$

In the given figure, AD = 3 cm, AE = 5 cm,

BD = 4 cm, CE = 4 cm, CF = 2 cm, BF = 2.5

cm, then find the pair of parallel lines and

hence their lengths.

$\\ \huge\bf\underline{Solution :-}$

In ∆ABC and ∆ EFC ,

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \angle \: ACB = \angle \: ECF \: \: - - - - - - (common \: angle)$

Now,

$\\ \sf \: \: \: \: \: \: \frac{CF }{CB} = \frac{2}{2 + 4.5} = \frac{2}{4.5} - - - - - (1)$

$\\ \sf \: \: \: \: \: \: \frac{CE }{CA} = \frac{4}{5+ 4} = \frac{4}{9} = \frac{2}{4.5} - - - - - (2)$

So, From 1 and 2 , we get

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \frac{CF }{CB} \: = \: \frac{CE }{CA}$

Since, Two corresponding sides are proportional and including angles are equal.

By SAS rule of similarity, we get

$\\ \\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ∆ABC \:≈ \: ∆ EFC$

So,

$\\ \sf \: \: \: \angle \: ABC = \angle \: ECF \: - - - (c.p.c.t.)$

$\\ \sf \: \: \: \angle \: BAC = \angle \: FCE \: - - - (c.p.c.t.)$

Since, the corresponding angles are equal therefore, By corresponding angle test we can say that line EF || Line AB .

So,

$\\ \sf \: \: \: \: \: \: \: \: \frac{EF }{AB} = \frac{2}{4.5}$

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: : \implies \: \frac{EF }{7} = \frac{2}{4.5}$

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: : \implies \: {EF }{} = \frac{2}{4.5} \times 7$

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: : \implies \: {EF }{} = \frac{14}{4.5}$

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: : \implies \: {EF }{} = \frac{14 \times 10}{45}$

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: : \implies \: {EF }{} = \frac{14 0}{45}$

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: : \implies \underline{ \boxed{ \orange{ \frak{ \: {EF }{} = 3.11}}}}$

Therefore,

The lines EF and AB are parallel. The length of EF is 3.11 cm.

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