If 4 cm and 7 cm are the lengths of two sides of a triangle, then the length of the third side may be……….
A. 11 cm
B. 3 cm
C. 6 cm
D. 12cm

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Hint: We’ll take the help of properties of the triangle to solve this question, applying those properties we’ll get some inequalities that will help us find the correct answer. Then we’ll look for the option which will best satisfy our inequalities to get the correct answer.

Complete step by step answer:

Given data: the length of two sides of the triangle is 4 cm and 7 cm.
Let the measure of third side of the triangle is ‘l’ cm
We know that the sum of two sides of a triangle is always greater than the measure of the third side.
We can say that,
$
  {\text{l < 4 + 7}} \\
   \Rightarrow {\text{l < 11cm}} \\
 $
It is also said that the measure of a side of a triangle is always greater than the difference of the other two sides of the triangle, concluding that
$
  {\text{l > }}\left| {{\text{7 - 4}}} \right| \\
   \Rightarrow {\text{l > 3cm}} \\
 $
From the above inequalities, we can say that ‘l’ lies between 3 cm and 11 cm.
From the options only (C)6 satisfy the above statement.
Hence, option (C) is the correct answer

Note: We can say that in a triangle always lies between the difference and the sum of the other two sides i.e., here
$
  \left| {{\text{7 - 4}}} \right|{\text{ < l < 7 + 4}} \\
   \Rightarrow {\text{3cm < l < 11cm}} \\
 $
Therefore, (C)6 cm