Question

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

1. 10, 10

10+10=20
10 - 10 = 0

1. 7, 11
2. 8, 6

8 - 6 = 2
2 < x < 14

1. 11, 8

Explanation:

Let's solve the problem for the sides 11 and 8 (problem 100).

Step 1: Recall the triangle inequality theorem

The triangle inequality theorem states that the length of any side of a triangle must be greater than the absolute difference of the other two sides and less than the sum of the other two sides. For two sides \(a\) and \(b\), the third side \(x\) must satisfy \(|a - b| < x < a + b\).

Step 2: Identify the given sides

Here, \(a = 11\) and \(b = 8\).

Step 3: Calculate the difference and sum

First, calculate the absolute difference: \(|11 - 8| = 3\).
Then, calculate the sum: \(11 + 8 = 19\).

Step 4: Determine the range for the third side

Using the triangle inequality theorem, the third side \(x\) must satisfy \(3 < x < 19\).

Answer:

The range of possible measures for the third side is \(3 < x < 19\).