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.

Step 1: 
Given that the two triangles \(\triangle AUV\) and \(\triangle ABC\) are similar, their corresponding sides are proportional. 
 

Step 2:

The proportion can be set up as follows: 
\[\frac { AU} { AB} = \frac { UV} { BC} = \frac { AV} { AC} \]

Step 3: 
Using the given lengths: 
\[\frac { AU} { AU + UB} = \frac { UV} { BC} = \frac { AV} { AC} \]

Step 4: 
Substitute the given values: 
\[\frac { 20x + 108} { 20x + 108 + 273} = \frac { 444} { 703} = \frac { 372} { 589} \]

Step 5: 
Simplify the equation for \(x\): 
\[\frac { 20x + 108} { 20x + 381} = \frac { 444} { 703} \]

Step 6: 
Cross-multiply to solve for \(x\): 
\[( 20x + 108) \times 703 = 444 \times ( 20x + 381) \]\[14060x + 75924 = 8880x + 169164\]

Step 7: 
Rearrange to isolate \(x\): 
\[14060x - 8880x = 169164 - 75924\]
\[5180x = 93240\]
\[x = \frac { 93240} { 5180} \]
\[x = 18\]
 

Supplemental Knowledge:

When two triangles are similar, their corresponding sides are proportional. This means that the ratios of the lengths of corresponding sides are equal.

Steps to Solve: 
1. Identify the corresponding sides of the similar triangles. 
2. Set up a proportion using the given side lengths. 
3. Solve for the unknown variable.

Real-Life Connections:

Solving problems involving triangles and proportions is crucial in real-life scenarios. Understanding this technique will assist with problem-solving for future endeavors.
Imagine yourself as an architect designing a scaled-down model of a building. Knowing how to apply proportions will allow for accuracy when replicating its full scaled counterpart in miniature form.

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