Question

two sides of a triangle have the following measures. find the range of possible measures for the third side.

1. 7, 12
2. 8, 7
3. 9, 8
4. 12, 6
5. 9, 10
6. 6, 9

classify each triangle by its angles and sides. equal sides and equal angles, if any, are

Explanation:

Step1: Apply the triangle inequality theorem

The triangle inequality theorem states that for a triangle with side lengths \(a\), \(b\), and \(c\), the following must hold: \(|a - b|\lt c\lt a + b\)

Step2: Solve for each problem

7) For sides \(a = 7\) and \(b=12\)

• Lower - bound: \(|12 - 7|=5\)
• Upper - bound: \(12 + 7 = 19\)

So, \(5\lt c\lt19\)

8) For sides \(a = 8\) and \(b = 7\)

• Lower - bound: \(|8 - 7|=1\)
• Upper - bound: \(8+7 = 15\)

So, \(1\lt c\lt15\)

9) For sides \(a = 9\) and \(b = 8\)

• Lower - bound: \(|9 - 8|=1\)
• Upper - bound: \(9 + 8=17\)

So, \(1\lt c\lt17\)

10) For sides \(a = 12\) and \(b = 6\)

• Lower - bound: \(|12 - 6|=6\)
• Upper - bound: \(12+6 = 18\)

So, \(6\lt c\lt18\)

11) For sides \(a = 9\) and \(b = 10\)

• Lower - bound: \(|10 - 9|=1\)
• Upper - bound: \(10 + 9=19\)

So, \(1\lt c\lt19\)

12) For sides \(a = 6\) and \(b = 9\)

• Lower - bound: \(|9 - 6|=3\)
• Upper - bound: \(9+6 = 15\)

So, \(3\lt c\lt15\)

Answer:

1. \(5\lt c\lt19\)
2. \(1\lt c\lt15\)
3. \(1\lt c\lt17\)
4. \(6\lt c\lt18\)
5. \(1\lt c\lt19\)
6. \(3\lt c\lt15\)