If the sides of a triangle are 7, 12, and x, and x is an integer, then find the number of possible values of x.
Show answer & explanation
Correct answer: A
Explanation
The correct answer is Option 1 i.e 13.
To determine the possible values of x, we use the triangle inequality theorem.
The theorem states that for any triangle with sides a, b, and c, the sum of the lengths of any two sides must be greater than the length of the third side.
For this triangle with sides 7, 12, and x:
⇒ 7 + 12 > x ⇒ x < 19
⇒ 7 + x > 12 ⇒ x > 5
⇒ 12 + x > 7 ⇒ x > -5 (this condition is always satisfied for positive x)
Thus, the values for x are constrained by
⇒ 5 < x < 19
Since x is an integer, the possible values of x are 6,7,8,…,18
So,
⇒ 18 - 6 + 1 = 13
So, there are 13 possible values for x