geometry side restriction of x

Question

Triangle ABC: side AB: not given. Side BC: x+10. Side AC: 2x+4. Angles: Angle A is greater than Angle B. What are the restrictions on x?

Answer 1

The larger the side, the larger the angle opposite that side. This is a consequence of the Law of Sines.

 

So since ∠B < ∠A, we have |AC| < |BC| or

 

2x + 4 < x + 10

 

x + 4 < 10

 

x < 6

 

Also, sides must have positive length so

 

2x + 4 > 0

 

2x > -4

 

x > -2

 

and also

 

x + 10 > 0

 

x > -10

 

Which means x > max(-2,-10) = -2

 

The constraint is thus

 

-2 < x < 6