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We are given that \( \triangle RST \) is isosceles with \( RS \cong TS \). This means that the lengths of sides \( RS \) and \( TS \) are equal. 

The expressions for the sides are as follows:
- \( RS = 9x + 2 \)
- \( TS = 6x + 8 \)
- \( RT = 3x - 1 \)

Since \( RS = TS \), we can set the two expressions equal to each other:
\[
9x + 2 = 6x + 8
\]
Now, solve for \( x \):

1. Subtract \( 6x \) from both sides:
\[
9x - 6x + 2 = 8
\]
\[
3x + 2 = 8
\]

2. Subtract 2 from both sides:
\[
3x = 6
\]

3. Divide by 3:
\[
x = 2
\]

Now that we know \( x = 2 \), substitute this value back into the expressions for \( RS \), \( TS \), and \( RT \) to find the lengths of each side:

1. \( RS = 9x + 2 = 9(2) + 2 = 18 + 2 = 20 \)
2. \( TS = 6x + 8 = 6(2) + 8 = 12 + 8 = 20 \)
3. \( RT = 3x - 1 = 3(2) - 1 = 6 - 1 = 5 \)

Thus, the correct answer is:
\[
x = 2, \quad RS = TS = 20, \quad RT = 5
\]
The correct option is **B**: \( x = 2, RS = TS = 20, RT = 5 \).

Would you like any further clarification or details?

Here are 5 related questions to deepen your understanding:
1. What are the properties of an isosceles triangle?
2. How would the solution change if the triangle were not isosceles?
3. Can the triangle inequality theorem be used to verify these side lengths?
4. What are some real-life examples where isosceles triangles are found?
5. How would the problem change if \( RT \) was given as equal to \( RS \) or \( TS \)?

**Tip:** When solving isosceles triangle problems, always start by equating the two equal sides and solving for unknowns, as the symmetry can simplify your work.

Find x and the length of each side if ΔRST is an isosceles triangle with RS ≅ TS.

Learn how to solve for x and find the side lengths of an isosceles triangle ΔRST where RS ≅ TS. This problem uses algebraic techniques to equate side lengths and solve linear equations. Suitable for Grades 7-9.

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