Question

Two sides of a ∆ have lengths 28cm and 82 cm. The measure of the third side is a whole number of centimeters. • What is the longest the third side can be? • What is the shortest the third side can be?

Answer 1

You need to remember the Triangl inequality Theorem. This states that

Let be "a", "b" and "c" the sides of a triangle. According to the Theorem mentioned above:

$\begin{gathered} a+b>c \\ b+c>a \\ a+c>b \end{gathered}$

In this case, knowing two sides of the triangle, you can set up that:

$\begin{gathered} a=28\operatorname{cm} \\ b=82\operatorname{cm} \end{gathered}$

Let be "c" the third side of this triangle. You know that:

$\begin{gathered} 28\operatorname{cm}+82\operatorname{cm}>c \\ 110\operatorname{cm}>c \end{gathered}$

Therefore, as you can notice, the third side can be less than 110 centimeters.

Based on the explained before, you can conclude that the third side can be:

$\begin{gathered} c<110\operatorname{cm} \\ \end{gathered}$

And it can be:

$\begin{gathered} c>82\operatorname{cm}-28\operatorname{cm} \\ c>54\operatorname{cm} \end{gathered}$

The answers are:

- The longest the third side can be is:

$109\operatorname{cm}$

- The shortest the third side can be is:

$55\operatorname{cm}$