The area of an isosceles trapezium is 176 cm2 and the height is 211^ th of the sum of its parallel sides. If the ratio of the length of the parallel sides is 4 : 7, then the length of a diagonal (in cm) is

Answer: Option A

Distance between two parallel line = $$\frac{2}{11}$$ (sum of both parallel line)
= $$\frac{2}{11}$$ × (7x + 4x)
= 2x
Area = $$\frac{1}{2}$$ (sum of parallel sides) × (distance between them)
⇒ $$\frac{1}{2}$$ (7x + 4x) × 2x = 176
⇒ 11x² = 176
⇒ x² = 16
⇒ x = 4
AB = 7 × 4 = 28 cm
CD = 4 × 4 = 16 cm
CM = 2 × 4 = 8 cm
AM = AN + NM
⇒ AM = AN + 16
⇒ AM = 6 + 16
⇒ AM = 22
$$\begin{array}{rl}& \left(\text{AN}=\text{BM}=\frac{12}{2}=6\right)\\ & \text{A}{\text{C}}^2=\text{C}{\text{M}}^2+\text{A}{\text{M}}^2\\ & \Rightarrow \text{A}{\text{C}}^2=8^2+22^2\\ & \Rightarrow \text{AC}=\sqrt{64+484}\\ & \Rightarrow \text{AC}=\sqrt{548}\\ & \Rightarrow \text{AC}=2\sqrt{137}\end{array}$$