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    A B C = 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 + 16 5 | = 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    P Q R = 0 1 2   | 2       6       3 k       1        k        9 2   2 1 | = 0 1 2 | 2 k + 27 + 3 k 6 3 k 2 9 | = 0 | 3 k 2 + 5 k + 12 | = 0 3 k 2 5 k 12 = 0 ( 3 k + 4 ) ( k 3 ) = 0                     k = 4 3   or   3

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