Question

two sides of a triangle have lengths 22 cm and 88 cm. the measure of the third side is a whole number of centimeters
a. what is the longest the third side can be?
b. what is the shortest the third side can be?
a. the longest the third side can be is
(simplify your answer.)

Explanation:

Step1: Apply triangle inequality theorem

The triangle inequality theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), \(a + b>c\), \(a + c>b\), and \(b + c>a\). Let the sides of the triangle be \(a = 22\) cm, \(b = 88\) cm, and \(c\) be the third - side. Then \(88−22 < c<88 + 22\).

Step2: Calculate the range of the third side

First, calculate \(88−22=66\) and \(88 + 22 = 110\). So \(66 < c<110\).

Step3: Find the longest value of the third side (for part a)

Since \(c\) is a whole number and \(c<110\), the longest value of \(c\) is \(109\) (because \(c\) must be less than \(110\) and an integer).

Step4: Find the shortest value of the third side (for part b)

Since \(c\) is a whole number and \(c>66\), the shortest value of \(c\) is \(67\) (because \(c\) must be greater than \(66\) and an integer).

Answer:

a. \(109\) cm
b. \(67\) cm