THE AREA OF A TRIANGLE 



27 



by measuring an angle 860 - a or n x 860 a in anti -clockwise 

 direction. 



Thus - 121 = 860 - 212 = 148 



- 589 = 720 - 589 - 181 



- 872 1080 - 872 == 108 



and the trigonometrical ratios of the corresponding positive 

 angles can be found in the usual way. 



15. The Area of a Triangle. If h and h l are the perpendiculars 

 drawn from A and C to the opposite sides respectively 



B 



CL 

 FIG. 9. 



Then area = - ah or - 



but 

 and 

 also 



- = sin B or h = c sin B 

 c 



r = sin C or h = b sin C 

 b 



= sin (180 - A) = sin A or Ji l = b sin A 



Hence area = - ac sin B = - ab sin C = - be sin A 



90 if 21 



Putting these relations in words, the area of a triangle is half 

 the product of two sides and the sine of the included angle. 



Let h be the perpendicular drawn from A, and x and (a x) 

 the segments into which the foot of the perpendicular divides 

 the base. 



Then h* = c* - a? 2 



and h* = ft 2 - (a - or) 2 

 b 2 - a 2 + 2ax - a? 2 = c 2 - a? 2 



2ax = a 2 + c* - b z 



2a 



