In a $Δ\ ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE\ ||\ BC$.
If $AD\ =\ 8x\ –\ 7\ cm$, $DB\ =\ 5x\ –\ 3\ cm$, $AE\ =\ 4x\ –\ 3\ cm$, and $EC\ =\ (3x\ –\ 1)\ cm$, Find the value of  $x$.

img src=/doubts_assets/images/158630-1605782187.png" style="width: 25%;">"

 Given:

In a $Δ\ ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE\ ||\ BC$.

$AD =8x-7 cm$, $AE =5x-3 cm$, $DB =4x-3 cm$ and $EC =3x-1 cm$.

To do:

We have to find the value of $x$.

Solution:

$DE\ ||\ BC$ (given)

Therefore,

By Basic proportionality theorem,

$\frac{AD}{DB}\ =\ \frac{AE}{EC}$

$ \begin{array}{l}
\frac{8x-7}{5x-3} =\frac{4x-3}{3x-1}\\
\\
( 8x-7)( 3x-1) =( 4x-3)( 5x-3)\\
\\
8x( 3x-1) -7( 3x-1) =4x( 5x-3) -3( 5x-3)\\
\\
24x^{2} -8x-21x+7=20x^{2} -12x-15x+9\\
\\
( 24-20) x^{2} +( -8-21+12+15) x+7-9=0\\
\\
4x^{2} -2x-2=0\\
\\
2x^{2} -x-1=0\\
\\
2x^{2} -2x+x-1=0\\
\\
2x( x-1) +1( x-1) =0\\
\\
( x-1)( 2x+1) =0\\
\\
x-1=0\ or\ 2x+1=0\\
\\
x=1\ or\ 2x=-1\\
\\
x=1\ or\ x=\frac{-1}{2}
\end{array}$

The side of a triangle cannot be negative. Therefore, the value of $x$ is $1 cm$. 

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Updated on: 10-Oct-2022

"&description=In a ABC D and E are points on the sides AB and AC respectively such that DE BC If AD 8x †7 cm DB 5x †3 cm AE 4x †3 cm and EC (3x †1) cm Find the value of x img src doubts assets images 158630 1605782187 png style width 25 -  Given:In a $Δ ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE || BC$.$AD =8x-7 cm$, $AE =5x-3 cm$, $DB =4x-3 cm$ and $EC =3x-1 cm$.To do:We have to find the value of $x$.Solution:$DE || BC$ (given)Therefore,By Basic proportionality theorem,$frac{AD}{DB} ','sharer','toolbar=0,status=0,width=580,height=325');" href="javascript: void(0)" title="Share on Facebook"> "&summary=In a ABC D and E are points on the sides AB and AC respectively such that DE BC If AD 8x †7 cm DB 5x †3 cm AE 4x †3 cm and EC (3x †1) cm Find the value of x img src doubts assets images 158630 1605782187 png style width 25 -  Given:In a $Δ ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE || BC$.$AD =8x-7 cm$, $AE =5x-3 cm$, $DB =4x-3 cm$ and $EC =3x-1 cm$.To do:We have to find the value of $x$.Solution:$DE || BC$ (given)Therefore,By Basic proportionality theorem,$frac{AD}{DB} ','sharer','toolbar=0,status=0,width=580,height=325');" href="javascript: void(0)" title="Share on LinkedIn">

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