Given a triangle ABC in which $ AB = x+3 $ and $ BC = x-1 $ and $\angle ABC = 30° $, its area is $ 35 \text{cm}^{2} $. Find $ AB+BC $.
+128474 Area = (1/2) ( AB) ( BC) sin 30
35 = (1/2) ( x + 3) ( x -1) (1/2
35 = (1/4)( x + 3) ( x -1) multiply through by 4
140 = (x + 3) ( x -1)
140 = x^2 + 2x - 3 rearrange as
x^2 + 2x -143 = 0 factor as
(x - 11) ( x + 13) = 0
The first factor set to 0 gives us what we need
x - 11 = 0
x =11
AB = x + 3 =14
BC = x - 1 = 10
AB + BC = 24
