Question

Triangle LMN is an isosceles triangle and UV with a line over it is a mix mid segment of the triangle the perimeter of the triangle LMN is 32NLM equals 3X find UV

Answer 1

In the context of an isosceles triangle \(LMN\) with a midsegment \(UV\), we can start by figuring out some details based on the given information.

1. **Perimeter of Triangle \(LMN\)**: Given that the perimeter is \(32\), we have:
\[
LM + LN + MN = 32
\]

2. **Understanding the midsegment \(UV\)**: The midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. The length of the midsegment is equal to half the length of the third side.

In the case of triangle \(LMN\):
- Assume \(MN\) is the base of the triangle.
- Let \(UV\) be the midsegment that connects the midpoints of sides \(LM\) and \(LN\).

Since \(UV\) is the midsegment:
\[
UV = \frac{1}{2} \cdot MN
\]

3. **Finding the Length of Side \(MN\)**: To find \(UV\), we first need to find the length of side \(MN\). We can express the sides in terms of \(x\) if we represent \(LM = LN = x\) and \(MN = y\). Thus:

\[
x + x + y = 32 \implies 2x + y = 32 \implies y = 32 - 2x
\]

4. **Calculate \(UV\)**: Substitute \(y\) into the equation for the midsegment:
\[
UV = \frac{1}{2}(MN) = \frac{1}{2}(y) = \frac{1}{2}(32 - 2x)
\]

Thus, simplifying further gives:
\[
UV = \frac{32 - 2x}{2} = 16 - x
\]

So, the final expression for the length of \(UV\) as a function of \(x\) is:
\[
UV = 16 - x
\]

If we know \(x\) (the length of each equal side), we could numerically evaluate \(UV\). However, since we do not have a specific value for \(x\), this is the general relationship.

### Conclusion
The length of the midsegment \(UV\) in reference to the equal length \(x\) of sides \(LM\) and \(LN\) is given by:
\[
UV = 16 - x
\]

If \(x\) is determined, substitute its value to find the exact length of \(UV\).