If $$ R V=x+6, U S=5 x-9 $$, and $$ R S=11 $$, find $$ V T $$ and $$ S T $$.

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To solve for $$ V T $$ and $$ S T $$, we first need to analyze the given information:

1. $$ R V = x + 6 $$
2. $$ U S = 5x - 9 $$
3. $$ R S = 11 $$

We can use the relationship between the segments to find $$ x $$. Since $$ R S $$ is the sum of $$ R V $$ and $$ V S $$, we can express $$ V S $$ as follows:

$$
R S = R V + V S
$$

This gives us:

$$
11 = (x + 6) + V S
$$

From this, we can solve for $$ V S $$:

$$
V S = 11 - (x + 6) = 11 - x - 6 = 5 - x
$$

Next, we can also express $$ R S $$ in terms of $$ U S $$ and $$ S T $$:

$$
R S = U S + S T
$$

Substituting $$ U S $$:

$$
11 = (5x - 9) + S T
$$

Now we can solve for $$ S T $$:

$$
S T = 11 - (5x - 9) = 11 - 5x + 9 = 20 - 5x
$$

Now we have two expressions:

1. $$ V S = 5 - x $$
2. $$ S T = 20 - 5x $$

Next, we need to find $$ x $$. We can use the fact that $$ R S = R V + V S $$ and substitute $$ V S $$:

$$
11 = (x + 6) + (5 - x)
$$

Simplifying this:

$$
11 = x + 6 + 5 - x
$$
$$
11 = 11
$$

This equation is always true, which means $$ x $$ can take any value. However, we can express $$ V T $$ and $$ S T $$ in terms of $$ x $$.

Now, we can find $$ V T $$ and $$ S T $$:

1. $$ V T = V S + S T = (5 - x) + (20 - 5x) = 25 - 6x $$
2. $$ S T = 20 - 5x $$

Thus, we have:

$$
V T = 25 - 6x
$$
$$
S T = 20 - 5x
$$

Since $$ x $$ can take any value, $$ V T $$ and $$ S T $$ are expressed in terms of $$ x $$. If you have a specific value for $$ x $$, you can substitute it in to find the numerical values of $$ V T $$ and $$ S T $$.
$$ V T = 25 - 6x $$ and $$ S T = 20 - 5x $$

If , and ,
find and .

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