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

The range of the possible measures for the third side of the triangle, given two side lengths of 35 and 12, is greater than 23 units and less than 47 units, according to the Triangle Inequality Theorem.

Step-by-step explanation:

To determine the range of possible measures for the third side of a triangle when two sides are given, one can use the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Given two sides of a triangle are 35 and 12 units, the third side must be greater than the difference between these two sides and less than their sum.

So, using this principle:

1. The third side must be greater than 35 - 12 = 23 units.
2. The third side must also be less than 35 + 12 = 47 units.

Therefore, the range for the measure of the third side is greater than 23 units but less than 47 units.

Answer 2

`\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}`