Question

Find all possible lengths of the third side, if sides of a triangle have $3$ and $9$.

• A. $6<x<12$
• B. $5<x<12$
• C. $6<x<10$
• D. $6<x<11$

Answer 1

The correct option is A $6<x<12$

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.
So, difference of two sides $<x<$ sum of two sides, will give you the possible length of a triangle.
Therefore, $9-3<x<9+3$
$6<x<12$ is the possible length of the third side of a triangle.
For checking the possible length: Take $3,9,7$
$3+9>7(a+b>c)$
$9+7>3(b+c>a)$
$3+7>9(a+c>b)$
Which satisfy the triangle inequality theorem.