6.3 Areas of Polygons

6.3 Areas of Polygons

(A) Area of Triangle


Area of Triangle



1. When the given points are taken in an anticlockwise direction the result is positive; taken in a clockwise direction the result is negative. The answer for the area must be given as a positive value.

Example 1:
Calculate the area of ∆ ABC with the vertices A (-5, 5), B (-2, -4), 
C (4, -1).

Solution:
Area of  ABC = 1 2 | 5   2      4    5     4   1   5   5 | = 1 2 | ( 5 )( 4 )+( 2 )( 1 )+( 4 )( 5 )       ( 5 )( 2 )( 4 )( 4 )( 1 ) ( 5 )| = 1 2 | 20+2+20+10+165| = 1 2 | 63| =31 1 2  unit 2

(B) Area of Quadrilateral



Area of Quadrilateral




(C) If the points A, B and C are collinear, then the area of ∆ ABC is 0.

Example 2:
Find the values of k if the points P (2, 1), Q (6, k) and R (3k, 9 2 ) are collinear.

Solution:
Area of  PQR=0 1 2  | 2     6      3k      1      k       9 2   2 1 |=0 1 2 | 2k+27+3k63 k 2 9|=0 | 3 k 2 +5k+ 12|=0 3 k 2 5k12=0 ( 3k+4 )( k3 )=0                    k= 4 3   or  3