Two sides of a triangle have lengths 6 and 16. Which inequalities represent the possible lengths for the third side, x?

## Which inequalities represent the possible lengths for the third side, x?

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# 3 Answers

let a,b, and c be the lengths of the three sides of a triangle

there are three inequalities:

a+b>c

a+c>b

b+c>a

for your lengths, let a=16 and b=6, therefore a+b>c or 16+6>c; so 22>c or c<22

c can be how much less than 22 ?

look at b+c>a, 6+c>16, c>10

therefore c can be in the following range: 10<c<22

*Rule for Δ ABC: any two sides added together must be greater than the third side.*

*If*

*AB = c*

*BC = a*

*AC = b,*

*then*

*a + b > c*

*a + c > b*

*b + c > a*

~~~~~~~~~~~~

If one side of triangle is 6 units, second side is 16 units and third one is "x" units then

x < 16 + 6

*x < 22*

6 + x > 16

*x > 10*

*10 < x < 22*

The sum of the lengths of any two sides of a triangle must be greater than the third side. Thus, in your case, any length greater than 16+6 = 22 is a valid length for x.