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The parallel sides of a trapezium are in ratio \[3:4\], if the distance between the parallel sides is \[9dm\] and its area is \[126d{{m}^{2}}\],find the length of its smallest side.

Answer Verified Verified
Hint:Here we will assume the length of the parallel sides first. In case of unknown terms assume any variable as the reference value. Then, use the formula for area of trapezium is equal to the half of the product of height with the sum of the length of its parallel sides and solve by substituting the known and unknown values to get the value of the assumed variable.

Complete step-by-step solution:
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 -
$\begin{align}
  & 3x=3\times 4=12dm\text{ } \\
 & \text{and 4x = 4}\times \text{4=16dm} \\
\end{align}$.

Hence the required solution is - the smallest segment will be 12dm.

Note: In these types of problems where dimensions are given in ratio, we first assume the ratio then form the equation using the required formula and put the assigned value given in question. Compare the assigned value and outcome of the equation and solve this will give us the required unknown value. Also, check the units of the values given and represent in the solution and final solution accordingly.