Question

Draw a quadrilateral in the Cartesian plane, whose vertices are (–4, 5), (0, 7), (5, –5) and (–4, –2). Also, find its area.

Answer 1

The vertices of a quadrilateral is given as A( −4,5 ), B( 0,7 ), C( 5,−5 ) and D( −4,−2 ).

Plot the given points on Cartesian plane and join AB, BC, CD and AD.

The quadrilateral formed after joining these points is shown below,

The above figure shows the quadrilateral formed by given vertices.

Draw a diagonal AC, then the area of the quadrilateral ABCD is,

area( ABCD )=area( ΔABC )+area( ΔACD )

It is known that the area of triangle with vertices ( x 1 , y 1 ), ( x 2 , y 2 ) and ( x 3 , y 3 ) is,

Area= 1 2 | x 1 ( y 2 − y 3 )+ x 2 ( y 3 − y 1 )+ x 3 ( y 1 − y 2 ) |

So, the area of ΔABC is,

area of ΔABC= 1 2 | −4( 7+5 )+0( −5−5 )+5( 5−7 ) | = 1 2 | −4( 12 )+5( −2 ) | = 1 2 | −58 | =29  unit 2

The area of ΔACD is,

area of ΔACD= 1 2 | −4( −5+2 )+5( −2−5 )+( −4 )( 5+5 ) | = 1 2 | −4( −3 )+5( −7 )−4( 10 ) | = 1 2 | −63 | = 63 2   unit 2

The area of the quadrilateral is,

area( ABCD )=29+ 63 2 = 58+63 2 = 121 2   unit 2

Therefore, the area of the quadrilateral ABCD is 121 2   unit 2 .