Question

The sum of the lengths of any two sides of a triangle is greater than the length of the third side. What can you conclude about AC in triangle ABC when BC=7 and AC=2+AB?

Answer 1

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side.

This rule must be satisfied for all 3 conditions of the sides

a + b > c

b + c > a

a + c > b

let AB = x

Given

BC = 7 ; a = 7 ,

AC = 2 + AB ; b = 2 + x ,

let AB = x ; c = x

Check 1st condition : 7 + 2 + x > x ( no solutions )

Check 3rd condition : 7+ x > 2 + x ( no solutions )

Check 2nd condition : 2 + x + x > 7   

=> 2 + 2x > 7

=> 2x > 5

=> x > 2.5

which implies AB > 2.5

so for AC,

AC = 2 + AB

as AB > 2.5

=> AC > 2 + 2.5  

=> AC > 4.5  

AC should be greater than 4.5 is the conclusion that can be drawn from the given information.