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Q.

If the coordinates of the vertices of a quadrilateral ABCD are A(-1, 6), B(-3, -9), C(5, -8), and D(3, 9), then what is its area?


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a

125 sq. units

b

126 sq. units

c

127 sq. units

d

128 sq. units 

answer is B.

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Detailed Solution

It is given that the coordinates of the vertices of a quadrilateral ABCD are A(-1, 6), B(-3, -9), C(5, -8), and D(3, 9).
In order to determine the area of a quadrilateral, we must first partition it into two triangles that do not share an area. We must then calculate the areas of these triangles and add them to determine the quadrilateral's area.
Question ImageFrom the figure, a quadrilateral ABCD is divided into triangles, ΔABC   and ΔADC.  
We know that if A x 1 , y 1 ,B x 2 , y 2 , and C x 3 , y 3   are the vertices of a triangle, then its area is given by,
ΔABC =  1 2 x 1 y 2 − y 3 + x 2 y 3 − y 1 + x 3 y 1 − y 2                       (1)  
⇒ΔABC= 1 2 [−1(−9−(−8))+(−3)(−8−6)+5(6−(−9))]  
⇒ΔABC= 1 2 [−1(−9−(−8))+(−3)(−8−6)+5(6−(−9))] ⇒ΔABC= 1 2 [1+72+75] ⇒ΔABC=74 sq. units  
and ΔACD= 1 2 [−1(−8−9)+5(9−6)+3(6−(−8))]  
⇒ΔACD= 1 2 [−1(−8−9)+5(9−6)+3(6−(−8))] ⇒ΔACD= 1 2 [17+15+72] ⇒ΔACD=52sq.units  
Therefore, the area of quadrilateral ABCD is,
⇒ar ΔABC +ar ΔACD ⇒ 74+52  sq. units ⇒126 sq. units  
Thus the area of the quadrilateral ABCD is 126 sq. units.
Hence, option (2) is correct.
 
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