Question

given lengths 19 and 14, determine which of the following values could or could not be the length of the third side of a triangle. select the correct option for each length.
length\ttriangle\tnot a triangle
2\t\t
18\t\t
3\t\t

Explanation:

To determine if a length can be the third side of a triangle with sides 19 and 14, we use the triangle inequality theorem, which states that the sum of any two sides must be greater than the third side, and the difference of any two sides must be less than the third side.

First, find the range of the third side \( x \):

The difference of the two given sides: \( 19 - 14 = 5 \)

The sum of the two given sides: \( 19 + 14 = 33 \)

So, the third side \( x \) must satisfy \( 5 < x < 33 \).

Step 1: Check length 2

Check if \( 5 < 2 < 33 \). Since \( 2 < 5 \), it does not satisfy the inequality. So, 2 is not a triangle.

Step 2: Check length 18

Check if \( 5 < 18 < 33 \). Since \( 5 < 18 \) and \( 18 < 33 \), it satisfies the inequality. So, 18 is a triangle.

Step 3: Check length 3

Check if \( 5 < 3 < 33 \). Since \( 3 < 5 \), it does not satisfy the inequality. So, 3 is not a triangle.

Answer:

• Length 2: Not a Triangle
• Length 18: Triangle
• Length 3: Not a Triangle