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 Given that the two triangles are similar, solve for x if AU = 20x + 108, UB = 273, BC = 703, UV = 444, AV = 372 and AC = 589.  You must show all of your work to receive credit.

To solve for \( x \), we use the fact that the triangles \( \triangle AUV \) and \( \triangle ABC \) are similar. This similarity implies that corresponding sides are proportional.

Given the sides:
- \( AU = 20x + 108 \)
- \( UB = 273 \)
- \( BC = 703 \)
- \( UV = 444 \)
- \( AV = 372 \)
- \( AC = 589 \)

The similarity of triangles gives us the following proportional relationships between corresponding sides:
\[ \frac{AU}{AB} = \frac{AV}{AC} = \frac{UV}{BC} \]

**Step 1: Calculate \( AB \)**
\[ AB = AU + UB = (20x + 108) + 273 = 20x + 381 \]

**Step 2: Set up the proportion using corresponding sides**
\[ \frac{AU}{AB} = \frac{AV}{AC} \]

Plug in the known values:
\[ \frac{20x + 108}{20x + 381} = \frac{372}{589} \]

**Step 3: Solve for \( x \)**
\[ (20x + 108) \cdot 589 = (20x + 381) \cdot 372 \]

Expanding both sides:
\[ 20x \cdot 589 + 108 \cdot 589 = 20x \cdot 372 + 381 \cdot 372 \]

\[ 11780x + 63612 = 7440x + 141732 \]

\[ 11780x - 7440x = 141732 - 63612 \]

\[ 4340x = 78000 \]

\[ x = \frac{78000}{4340} \]

\[ x \approx 17.97 \]

Thus, the value of \( x \) is approximately \( 17.97 \).

Would you like a further breakdown of these steps or have any other questions? Here are some related questions to explore:

1. How can you verify if two triangles are similar?
2. What are the criteria for two triangles to be similar?
3. How does the concept of similarity help in solving geometric problems?
4. What are other methods to find the missing sides in similar triangles?
5. How can the proportionality of corresponding sides be used in real-world applications?
6. What is the significance of the similarity ratio in similar triangles?
7. How do you solve problems involving similar triangles with more than one unknown?
8. What are some common mistakes to avoid when working with similar triangles?

**Tip:** When working with similar triangles, always ensure that the corresponding sides are correctly identified and that the proportions are set up accurately. This ensures the correct application of the similarity ratio.

Learn how to solve for x in a problem involving similar triangles (AU, UB, BC, UV, AV, AC). Understand the concept of triangle similarity and use proportional relationships to find the value of x step-by-step. Suitable for high school students studying geometry.

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