question_answer

The area of a trapezium with equal non- parallel sides is\[\text{168 }{{\text{m}}^{\text{2}}}\]. If the lengths of the parallel sides are 36 m and 20 m, find the length of the non-parallel sides.

A)  8m                             

B)  10m   

C)         15m   

D)         4m

Correct Answer: B

Solution :

Let the length of non-parallel sides AD and BC be x m. Here AB||CD and AB=20m, CD=36m Area of trapezium \[ABCD=168{{m}^{2}}\] Let the distance between two parallel sides be \[h\]. \[\therefore \]\[168=\frac{1}{2}\times (AB+CD)\times h\] \[168=\frac{1}{2}\times 56\times h\Rightarrow h=6m\] Now draw \[\text{BE }\!\!|\!\!\text{  }\!\!|\!\!\text{ AD}\] Given, \[\text{AB }\!\!|\!\!\text{  }\!\!|\!\!\text{ CD}\Rightarrow \text{AB }\!\!|\!\!\text{  }\!\!|\!\!\text{ DE}\] \[\therefore \] ABED is a parallelogram. \[\Rightarrow BE=AD=x\] and \[AB=DE=20m\] So, \[EC=36-20=16m\] In \[\Delta BEC,\] we have \[BE=BC=x\,\,m\] \[\Rightarrow \Delta BEC\] is an isosceles triangle. \[\therefore \]\[EM=MC=\frac{1}{2}EC=\frac{1}{2}\times 16=8m\] By Pythagoras theorem in \[\Delta BEM,\] \[B{{E}^{2}}=B{{M}^{2}}+E{{M}^{2}}\] \[\Rightarrow {{x}^{2}}={{6}^{2}}+{{8}^{2}}=100\Rightarrow x=10m\] Hence, the length of non-parallel side is 10m.