5.2 Area of a Triangle

. Area of a Triangle In your earlier classes, you have studied how to calculate the area of a triangle when its base and corresponding height (altitude) are given. You have used the formula. Area of triangle = × base × altitude sq.units. With any three non-collinear points A x y ( , , B x y ( , and C y y ( , on a plane, we can form a triangle ABC . Using distance between two points formula, we can calculate AB , BC CA = . a , b , c represent the lengths of the sides of the triangle ABC . Using s + , we can calculate the area of triangle ABC by using the Heron’s formula s s a s b s . But this procedure of finding length of sides of D ABC and then calculating its area will be a tedious procedure. There is an elegant way of finding area of a triangle using the coordinates of its vertices. We shall discuss such a method below. Let ABC be any triangle whose vertices are at A x y ( , , B x y ( , and C x y . Draw AP, BQ and CR perpendiculars from A, B and C to the x -axis, respectively. Clearly ABQP , APRC and BQRC are all trapeziums. ( x , y ) ( x , y ) ( x , y ) Fig. . X X ′ ¢ C ( x ,y ) A ( x , y ) ( x , y ) Q P R X X ′ ¢ Fig. . Now from Fig. . , it is clear that Area of D ABC = Area of trapezium ABQP + Area of trapezium APRC − Area of trapezium BQRC . You also know that, the area of trapezium (sum of parallel sides) ´ (perpendicular distance between the parallel sides) Therefore, Area of D ABC BQ AP QP AP CR PR BQ CR QR