Question

$ABCD$ is a trapezium with $AB\parallel CD$ and its area is $33c{m}^2.$ From the information given in the figure, find the lengths of all sides of the $ABCD.$

Answer 1

Given: $AB\parallel CD$

Area of a trapezium $=\frac{1}{2}\times$ (sum of parallel sides) $\times$ (height)

$=\frac{1}{2}\times (x+2x+1)(x-4)$

$=\frac{1}{2}(3x+1)(x-4)$

$=\frac{{3x}^2-12x+x-4}{2}$

$=\frac{{3x}^2-11x-4}{2}$

Given: the area of trapezium is ${33cm}^2$

$\Rightarrow \frac{{3x}^2-11x-4}{2}=33$

$\Rightarrow {3x}^2-11x-4=66$

$\Rightarrow {3x}^2-11x-70=0$

$\Rightarrow {3x}^2-21x+10x-70=0$

$\Rightarrow 3x(x-7)+10(x-7)=0$

$\Rightarrow (3x+10)(x-7)=0$

$\therefore$ $x=7$ units

Hence the sides of the trapezium are:

$\left|AB\right|=x=7cm$

$\left|BC\right|=x-2=5cm$

$\left|CD\right|=2x+1=11cm$

$\left|AD\right|=\left|BC\right|=5cm$