Question

Two sides of a triangle have the measures 35 and 12. Find the range of the possible measures for the third side of the triangle.

Answer 1

ANSWER

$\text{23 }\leq\text{ x }\leq\text{ 47}$

EXPLANATION

We have that the two sides of the triangle are 35 and 12.

To find the possible range of values for the third side, we have to apply the Triangle Inequality Theorem.

It states that the sum of two sides of a triangle must always be greater than the third side.

Let the third side be x.

This could mean three things:

$\begin{gathered} x\text{ + 12 }\ge\text{ 35} \\ or\text{ } \\ x\text{ + 35 }\ge12 \\ or \\ 12\text{ + 35 }\ge x \end{gathered}$

We will simplify each of them to see the possible range:

$\begin{gathered} \Rightarrow x\text{ }\ge35\text{ - 12} \\ x\text{ }\ge23 \\ \Rightarrow x\text{ }\ge12\text{ - 35} \\ x\text{ }\ge-23 \\ \Rightarrow\text{ 12 + 35 }\ge x \\ 47\text{ }\ge x\text{ or x }\leq47 \end{gathered}$

The second inequality is invalid because the side of a triangle cannot be negative.

So, the possible range of values of x is:

$\begin{gathered} x\text{ }\ge\text{ 23 and x }\leq\text{ 47} \\ \Rightarrow\text{ 23 }\leq\text{ x }\leq\text{ 47} \end{gathered}$