Question

given two sides of a triangle, determine the range of possible lengths of the third side. see example 5.

1. 10 in. and 12 in.

1. 5 ft and 10 ft

1. 200 m and 300 m

1. 90 km and 150 km

Explanation:

Step1: Determine the difference and sum of the two given sides

For a triangle with sides \(a\) and \(b\), the length of the third side \(c\) must satisfy the inequality \(|a - b|\lt c\lt a + b\).
For the sides \(a = 10\) in and \(b=12\) in:
The difference is \(|10 - 12|=2\) in.
The sum is \(10 + 12 = 22\) in.

Step2: Write the range of the third - side length

Based on the triangle - inequality theorem, if \(a = 10\) in and \(b = 12\) in, then \(2\lt c\lt22\).

For the sides \(a = 5\) ft and \(b = 10\) ft:
The difference is \(|5 - 10|=5\) ft.
The sum is \(5+10 = 15\) ft.
So, \(5\lt c\lt15\).

For the sides \(a = 200\) m and \(b = 300\) m:
The difference is \(|200 - 300|=100\) m.
The sum is \(200 + 300=500\) m.
So, \(100\lt c\lt500\).

For the sides \(a = 90\) km and \(b = 150\) km:
The difference is \(|90 - 150|=60\) km.
The sum is \(90+150 = 240\) km.
So, \(60\lt c\lt240\).

Answer:

1. The range of the third - side length is \(2\lt c\lt22\) (in inches).
2. The range of the third - side length is \(5\lt c\lt15\) (in feet).
3. The range of the third - side length is \(100\lt c\lt500\) (in meters).
4. The range of the third - side length is \(60\lt c\lt240\) (in kilometers).