Given, three segments of length x, (11 -x) and (x - 4) respectively. Which of the following indicates the set of all numbers "x" such that the three segments could be the lengths of the side of a triangle?
options:
(A) x>4
(B) 0<x<11
(C)4<x<11
(D)5<x<7

Answer :

Given :  three segments of length x, (11 - x ) and ( x  - 4 ) represent As length of sides of triangle .

We know from triangle inequality , Sum of any two sides of triangle is always greater than the length of third side of triangle .

So,

x < 11 - x + x - 4

$\Rightarrow$x < 7                                       ----------- ( 1 )

And

x  - 4 < x  + 11 - x

$\Rightarrow$x - 4 < 11

$\Rightarrow$x < 15                                       ----------- ( 2 )
And

11 - x < x + x - 4

$\Rightarrow$2x - 4 > 11 - x

$\Rightarrow$3x > 15

$\Rightarrow$x  > 5                                       ----------- ( 3 ) 

SO,
From equation 1 , 2 and 3 , we get

5 < x  < 7

Option ( D )                                                                 ( Ans )