Two sides of a plot measure 32 meters and 24 meters and the angle between them is a perfect right angle. The other two sides measure 25 meters each and the other three angles are not right angles. Find the area of quadrilateral (in sq. meters).

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Solution

$W e n a m e a n d r e c o n s t r u c t t h e f i g u r e a s s h o w n . G i v e n - A B C D i s a f i e l d w i t h A B = 32 m, B C = C D = 25 m a n d A D = 24 m . ∠ A = 90^{o} a n d n o o t h e r a n g l e i s 90^{o} . T o f i n d o u t - a r . A B C D = ? S o l u t i o n - Δ A B D i s a r i g h t t r i a n g l e s i n c e ∠ A = 90^{o} . ∴ B D = \sqrt{{A D}^{2} + {A B}^{2}} = \sqrt{24^{2} + 32^{2}} m = 40 m . N o w Δ B D C i s i s o c e l e s a s D C = 25 m = B C . S o i t s h e i g h t C E w i l l . m e e t B D a t E w h i c h i s t h e m i d p o i n t o f B D a n d ∠ B E C = 90^{o} i . e Δ B E C i s a r i g h t t r i a n g l e . ∴ B E = \frac{1}{2} B D = \frac{1}{2} \times 40 m = 20 m a n d C E = \sqrt{{B C}^{2} - {B E}^{2}} = \sqrt{25^{2} - 20^{2}} m = 15 m . ∴ a r Δ A B D = \frac{1}{2} \times A B \times A D = \frac{1}{2} \times 32 \times 24 s q . m = 384 s q . m a n d a r Δ B D C = \frac{1}{2} \times D B \times C E = \frac{1}{2} \times 40 \times 15 s q . m = 300 s q . m . N o w a r . A B C D = a r . Δ A B D + a r . Δ B D C = (384 + 300) s q . m = 684 s q . m . A n s - a r . A B C D = 684 s q . m .$

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