Question

1. find the minimum and maximum limits for the length of a third side of a triangle if the other two sides are 8\ and 13\.

Explanation:

Step1: Recall triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side, and the difference between the lengths of any two sides must be less than the length of the third side. Let the two given side - lengths be \(a = 8\) and \(b = 13\), and the third side be \(x\).

Step2: Calculate the lower - bound

By the triangle - inequality \(|a - b|\lt x\). So, \(|8 - 13|=| - 5| = 5\), and \(x>5\).

Step3: Calculate the upper - bound

By the triangle - inequality \(x\lt a + b\). So, \(x<8 + 13=21\).

Answer:

The minimum limit for the length of the third side is greater than 5, and the maximum limit is less than 21.