Question
Tue March 17, 2015 By: Sthitaprajna Mishra

To find the area of a quadrilateral we divide it into two triangles and then find the area of each triangle with the help of the triangle formula and add them. But I have noticed a pattern in them. For ex:
 
Find the area of the quadrilateral formed by joining the points - A(-4,-2), B(-3,-5), C(3,-2), D(2,3).
 
In questions like this, we get the area correctly only if we add the areas of traingles ABC and ACD. We don't get the correct answer if we add them in a different manner, say triangle ABC and ADC etc. We can even check this on the above question. We get the answer only if add the triangles ABC and ACD and not in some other way. Is there a reason for this? If yes, please explain.

Expert Reply
Vimala Ramamurthy
Wed March 18, 2015
C o n s i d e r space t h e space a r e a space o f space t h e space t r i a n g l e space A B C colon A open parentheses triangle A B C close parentheses equals 1 half open square brackets open parentheses negative 4 close parentheses open parentheses negative 5 plus 2 close parentheses minus 3 open parentheses negative 2 plus 2 close parentheses plus 3 open parentheses negative 2 plus 5 close parentheses close square brackets space space space space space space space space space space space space space space space space equals 1 half open square brackets 12 plus 9 close square brackets equals 21 over 2 space s q u a r e space u n i t s. C o n s i d e r space t h e space a r e a space o f space t h e space t r i a n g l e space A C D colon A open parentheses triangle A C D close parentheses equals 1 half open square brackets open parentheses negative 4 close parentheses open parentheses negative 2 minus 3 close parentheses plus 3 open parentheses 3 plus 2 close parentheses plus 2 open parentheses negative 2 plus 2 close parentheses close square brackets space space space space space space space space space space space space space space space space equals 1 half open square brackets 20 plus 15 close square brackets equals 35 over 2 space s q u a r e space u n i t s. T o t a l space a r e a equals A open parentheses triangle A B C close parentheses plus A open parentheses triangle A C D close parentheses equals 21 over 2 plus 35 over 2 equals 56 over 2 equals 28 space s q u a r e space u n i t s. N o w space c o n s i d e r space t h e space a r e a space o f space t h e space t r i a n g l e space A D C colon A open parentheses triangle A D C close parentheses equals 1 half open square brackets open parentheses negative 4 close parentheses open parentheses 3 plus 2 close parentheses plus 2 open parentheses negative 2 plus 2 close parentheses plus 3 open parentheses negative 2 minus 3 close parentheses close square brackets space space space space space space space space space space space space space space space space equals 1 half open square brackets negative 20 minus 15 close square brackets equals open vertical bar negative 35 over 2 close vertical bar equals 35 over 2 space s q u a r e space u n i t s. S i n c e space a r e a space c a n n o t space b e space n e g a t i v e space w e space a r e space c o n s i d e r i n g space t h e space a b s o l u t e space v a l u e. T h u s comma space T o t a l space a r e a equals A open parentheses triangle A B C close parentheses plus A open parentheses triangle A D C close parentheses equals 21 over 2 plus 35 over 2 equals 56 over 2 equals 28 space s q u a r e space u n i t s. I n space b o t h space t h e space w a y s space t h e space a n s w e r s space a r e space s a m e.
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