Question

In figure DE||BC, AD=2.4cm, AE=3.2cm, CE=4.8cm. the value of BD is

$A.$ 3.6cm
$B.$ 4.2cm
$C.$ 4.0cm
D. none of these

Answer 1

Hint: In this question, we will use the concept of similarity of triangles. Two triangles are said to be similar if the corresponding sides of the two triangles are proportional to each other. By making similar two triangles ADE and ABC we find the solution of the given problem.

Complete step by step answer:
Now moving to the solution
We have, in triangles ABC and ADE,
$\angle A = \angle A$ (common angle)
$\angle ADE = \angle ABC$ ( corresponding angles, as DE||BC)
$\therefore \vartriangle ADE \sim \vartriangle ABC$ ( by AA criteria)
Hence, $\dfrac{{AD}}{{AB}} = \dfrac{{AE}}{{AC}}$ .…..(1)
Let, BD = x
AC = AE + CE
AC = 3.2 + 4.8
AC = 8cm
AB = AD + BD
AB = 2.4 + x
Substituting these values in equation (1), we get
$\dfrac{{2.4}}{{2.4 + x}} = \dfrac{{3.2}}{8}$
$19.2 = 7.68 + 3.2x$
$x = \dfrac{{11.52}}{{3.2}}$
$x = 3.6cm$
Which means option $A.$ is the correct option.
Note: Another method – by basic proportionality theorem
 It states that if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio. This is also known as BPT.
By basic proportionality theorem,
$\dfrac{{AD}}{{DB}} = \dfrac{{AE}}{{EC}}$
$\dfrac{{2.4}}{{DB}} = \dfrac{{3.2}}{{4.8}}$
$ \Rightarrow \dfrac{{2.4}}{{DB}} = \dfrac{2}{3}$
$ \Rightarrow DB = \dfrac{{2.4 \times 3}}{2}$
$ \Rightarrow DB = 3.6cm$or $BD = 3.6cm$