Question

given two sides of a triangle, find a range of possible lengths for the third side
10 6. 12 x
range
11 8, 13, x
range
12 12. 17 x
13 8, 12, x

Explanation:

Step1: Apply the triangle - side - length rule

For a triangle with side lengths \(a\), \(b\), and \(c\), the following inequalities hold: \(|a - b|\lt c\lt a + b\)

For the first set of sides \(6\), \(12\)

• Calculate \(|12 - 6|\)

\(|12 - 6|=6\)

• Calculate \(12 + 6\)

\(12+6 = 18\)
So, \(6\lt x\lt18\)

For the second set of sides \(8\), \(13\)

• Calculate \(|13 - 8|\)

\(|13 - 8| = 5\)

• Calculate \(13+8\)

\(13 + 8=21\)
So, \(5\lt x\lt21\)

For the third set of sides \(12\), \(17\)

• Calculate \(|17 - 12|\)

\(|17 - 12|=5\)

• Calculate \(17+12\)

\(17 + 12=29\)
So, \(5\lt x\lt29\)

For the fourth set of sides \(8\), \(12\)

• Calculate \(|12 - 8|\)

\(|12 - 8| = 4\)

• Calculate \(12+8\)

\(12 + 8=20\)
So, \(4\lt x\lt20\)

Answer:

1. \(6\lt x\lt18\)
2. \(5\lt x\lt21\)
3. \(5\lt x\lt29\)
4. \(4\lt x\lt20\)