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Given: Ratio of parallel sides be \[3:4\]and area \[126d{{m}^{2}}\]with distance between parallel sides \[9dm\].

Let us suppose that the common factor in the ratio is \[x\].

Its parallel sides will be\[3x\text{ }and\text{ }4x\].

Distance between two parallel sides is \[9dm\](i.e measure of the height)

Now, Area of trapezium = $A=\dfrac{height}{2}\times (sum\text{ }of\text{ the length of the }sides)$

Put known and unknown values in the above equation -

$\begin{align}

& \Rightarrow 126=\dfrac{9(3x+4x)}{2} \\

& \Rightarrow 126=\dfrac{9(7x)}{2} \\

\end{align}$

Simplify by using cross – multiplication, when the term in the denominator changes its side it is multiplied with the numerator in the opposite side

Lengths of the sides are -

$\begin{align}

& 3x=3\times 4=12dm\text{ } \\

& \text{and 4x = 4}\times \text{4=16dm} \\

\end{align}$.