triangle vwx is formed by connecting the midpoints of the side of triangle stu. the lengths of the sides of triangle vwx are shown. find the perimeter of triangle stu. figures not necessarily drawn to scale.

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# Explanation:
## Step1: Recall Midsegment Theorem
The Midsegment Theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long. In triangle $ STU $, $ V, W, X $ are midpoints, so $ VW $, $ WX $, $ VX $ are midsegments of $ \triangle STU $.

## Step2: Determine side lengths of $ \triangle STU $
- For side $ ST $: Since $ WX $ is a midsegment, $ ST = 2 \times WX $. Given $ WX = 2 $, so $ ST = 2\times2 = 4 $.
- For side $ TU $: Since $ VX $ is a midsegment, $ TU = 2 \times VX $. Given $ VX = 2 $, so $ TU = 2\times2 = 4 $.
- For side $ SU $: Since $ VW $ is a midsegment, $ SU = 2 \times VW $. Given $ VW = 2 $, so $ SU = 2\times2 = 4 $. Wait, no, wait. Wait, actually, looking at the triangle $ VWX $, each side is 2. Wait, maybe I misread. Wait, triangle $ VWX $ has sides $ VX = 2 $, $ WX = 2 $, $ VW = 2 $? Wait, no, the diagram: $ X $ is midpoint of $ SU $, $ W $ is midpoint of $ TU $, $ V $ is midpoint of $ ST $. So $ VX $ connects midpoints of $ ST $ and $ SU $, so $ VX \parallel TU $ and $ VX = \frac{1}{2}TU $. Similarly, $ WX \parallel ST $ and $ WX = \frac{1}{2}ST $, $ VW \parallel SU $ and $ VW = \frac{1}{2}SU $.

Wait, the perimeter of $ \triangle VWX $ is $ 2 + 2 + 2 = 6 $? No, wait the problem is to find perimeter of $ \triangle STU $. Wait, no, wait the question: "Find the perimeter of triangle $ STU $". Wait, triangle $ VWX $ is formed by midpoints, so by Midsegment Theorem, each side of $ \triangle STU $ is twice the corresponding side of $ \triangle VWX $.

Looking at the diagram, $ \triangle VWX $ has sides $ VX = 2 $, $ WX = 2 $, $ VW = 2 $? Wait, no, maybe each side of $ \triangle VWX $ is 2, so each side of $ \triangle STU $ is $ 2 \times 2 = 4 $. Wait, but let's check again.

Wait, the triangle $ VWX $: $ X $ is midpoint of $ SU $, $ W $ is midpoint of $ TU $, $ V $ is midpoint of $ ST $. So $ VX $ is midsegment of $ \triangle STU $, so $ VX \parallel TU $ and $ VX = \frac{1}{2}TU $. So $ TU = 2 \times VX = 2\times2 = 4 $. Similarly, $ WX \parallel ST $, so $ ST = 2 \times WX = 2\times2 = 4 $. $ VW \parallel SU $, so $ SU = 2 \times VW = 2\times2 = 4 $. Wait, so $ \triangle STU $ is equilateral with each side 4? Then perimeter is $ 4 + 4 + 4 = 12 $? Wait, no, wait, maybe I made a mistake. Wait, no, the sides of $ \triangle VWX $: if $ VX = 2 $, $ WX = 2 $, $ VW = 2 $, then each side of $ \triangle STU $ is $ 2 \times 2 = 4 $, so perimeter is $ 4 + 4 + 4 = 12 $. Wait, but let's re-express:

Perimeter of $ \triangle STU = ST + TU + SU $.

By Midsegment Theorem:

- $ ST = 2 \times WX $ (since $ WX $ is midsegment between $ S $ and $ U $? Wait, no, $ W $ is midpoint of $ TU $, $ X $ is midpoint of $ SU $, so $ WX $ is midsegment of $ \triangle STU $, so $ WX \parallel ST $ and $ WX = \frac{1}{2}ST $, so $ ST = 2 \times WX = 2\times2 = 4 $.

- $ TU = 2 \times VX $ ( $ V $ is midpoint of $ ST $, $ X $ is midpoint of $ SU $, so $ VX $ is midsegment, $ VX \parallel TU $, $ VX = \frac{1}{2}TU $, so $ TU = 2\times VX = 2\times2 = 4 $.

- $ SU = 2 \times VW $ ( $ V $ is midpoint of $ ST $, $ W $ is midpoint of $ TU $, so $ VW $ is midsegment, $ VW \parallel SU $, $ VW = \frac{1}{2}SU $, so $ SU = 2\times VW = 2\times2 = 4 $.

Thus, perimeter of $ \triangle STU = 4 + 4 + 4 = 12 $. Wait, but that seems too simple. Wait, maybe the sides of $ \triangle VWX $ are each 2, so perimeter of $ \triangle STU $ is $ 2\times(2 + 2 + 2) = 12 $. Yes, because each side of $ STU $ is twice each side of $ VWX $, so perimeter of $ STU $ is twice perimeter of $ VWX $. Perimeter of $ VWX $ is $ 2 + 2 + 2 = 6 $, so perimeter of $ STU $ is $ 2\times6 = 12 $.

# Answer:
12

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Question

Explanation:

Step1: Recall Midsegment Theorem

Step2: Determine side lengths of \( \triangle STU \)

Answer: