The vertices of a quadrilateral are (2, -1), (3, 4), (-2, 3) and (-3, -2). Find it's area without using the area formula.

Asked by Simran K | 23rd Feb, 2014, 01:24: PM

Expert Answer:

Consider the vertices, A(2, -1), B(3, 4), C(-2, 3) and D(-3, -2).
 
Compute, d1 = AB = square root of open parentheses 3 minus 2 close parentheses squared plus open parentheses 4 plus 1 close parentheses squared end root equals square root of 26
d2 = BC = square root of open parentheses minus 2 minus 3 close parentheses squared plus open parentheses 3 minus 4 close parentheses squared end root equals square root of 26
d3 = CD = square root of open parentheses minus 3 plus 2 close parentheses squared plus open parentheses minus 2 minus 3 close parentheses squared end root equals square root of 26
d4 = DA = square root of open parentheses minus 3 minus 2 close parentheses squared plus open parentheses minus 2 plus 1 close parentheses squared end root equals square root of 26
Thus, the sides AB = BC= CD = DA = square root of 26.
Hence the quadrilateral is the rhombus.
 
Area of the rhombus = 1 half cross times D subscript 1 cross times D subscript 2 comma space w h e r e comma space D subscript 1 space a n d space D subscript 2 space end subscript a r e space t h e space d i a g o n a l s space o f space t h e space r h o m b u s
Compute D subscript 1 equals square root of open parentheses 3 plus 3 close parentheses squared plus open parentheses 4 plus 2 close parentheses squared end root equals square root of 36 plus 36 end root equals 6 square root of 2
and D subscript 2 equals square root of open parentheses 2 plus 2 close parentheses squared plus open parentheses minus 1 minus 3 close parentheses squared end root equals square root of 16 plus 16 end root equals 4 square root of 2
Thus, the area of the rhombus is A equals 1 half cross times 6 square root of 2 cross times 4 square root of 2 equals 24 space s q. u n i t s

Answered by  | 25th Feb, 2014, 11:14: AM

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