question_answer

The parallel sides of a trapezoid are \[3\,cm\] and \[9\,cm\]. The non-parallel sides are \[4\,cm\] and \[6\,cm\]. A line parallel to the base divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided, is:

A) \[4:5\]

B) \[3:2\]     

C) \[4:1\]

D) \[3:1\]

Correct Answer: C

Solution :

[c]

\[A.T.Q\,\,\,x+3+y+z=9+6-y+z+4-x\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,x+3+y=19-x-y\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,2x+2y=16\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,x+y=8\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,y=8-x\]

Also      \[\frac{x}{4-x}=\frac{y}{6-y}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{x}{4-x}=\frac{8-x}{6-8+x}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,x(x-2)=(8-x)\,\,(4-x)\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,{{x}^{2}}-2x=32-12x+{{x}^{2}}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,x=3.2cm\]

\[\therefore \]      Ratio \[=\frac{3.2}{4-3.2}=\frac{4}{1}\]