The lengths of parallel sides of a trapezium are (x + 9) cm and (2x - 3) cm , and the distance between them is (x + 4) cm . If its area is 540 cm² , find x .

Given ,

Length of first parallel side = (x + 9) cm

Length of second parallel side = (2x - 3) cm

Distance between parallel side = (x + 4) cm

Area of trapezium = 540 cm²

Area of trapezium is given by,

$=\dfrac{1}{2} \times \text{ (sum of parallel sides) } \times \text{ (distance between them) } \\[1em] \Rightarrow \dfrac{1}{2} \times (x + 9 + 2x - 3) \times (x + 4) = 540 \\[1em] \Rightarrow \dfrac{1}{2} \times (3x + 6)(x + 4) = 540 \\[1em] \Rightarrow \dfrac{1}{2} \times (3x^2 + 12x + 6x + 24) = 540 \\[1em] \Rightarrow \dfrac{1}{2} \times (3x^2 + 18x + 24) = 540 \\[1em] \Rightarrow 3x^2 + 18x + 24 = 540 \times 2 \text{ (On cross multiplication) } \\[1em] \Rightarrow 3x^2 + 18x + 24 = 1080 \\[1em] \Rightarrow 3x^2 + 18x + 24 - 1080 = 0 \\[1em] \Rightarrow 3x^2 + 18x - 1056 = 0 \\[1em] \Rightarrow 3(x^2 + 6x - 352) = 0 \\[1em] \Rightarrow x^2 + 6x - 352 = 0 \\[1em] \Rightarrow x^2 + 22x - 16x - 352 = 0 \\[1em] \Rightarrow x(x + 22) - 16(x + 22) = 0 \\[1em] \Rightarrow (x - 16)(x + 22) = 0 \\[1em] \Rightarrow x - 16 = 0 \text{ or } x + 22 = 0 \\[1em] x = 16 \text{ or } x = -22$

If x = -22 , Length = x + 9 = -22 + 9 = -13 , Breadth = (2x - 3) = -44 - 3 = -47
Since length and breadth cannot be negative hence , x ≠ -22

∴ x = 16

The value of x is 16.