1. Identify Given Information: We are given a figure with triangle RST and points A on RS, B on RT.
\( RA = 6 \)
\( AS = 2 \)
\( AB = 9 \)
\( BT = 2 \)
\( ST = x \)
The triangles are similar: \( \Delta RST \sim \Delta RBA \)
From the segment lengths, we can find the length of side RS: \( RS = RA + AS = 6 + 2 = 8 \)
2. Use the Property of Similar Triangles: Since \( \Delta RST \sim \Delta RBA \), the ratio of corresponding sides must be equal. The correspondence of vertices is R to R, S to B, and T to A. Therefore, the ratios of corresponding sides are: $$ \frac{RS}{RB} = \frac{ST}{BA} = \frac{RT}{RA} $$
3. Determine Unknown Side Lengths (RB and RT): The arrows on segments AB and ST in the figure indicate that \( AB \parallel ST \). When a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally (by Thales' Theorem or Basic Proportionality Theorem, and properties of similar triangles \( \Delta RAB \sim \Delta RST \)). Thus, we have: $$ \frac{RA}{AS} = \frac{RB}{BT} $$ Substituting the known values: $$ \frac{6}{2} = \frac{RB}{2} $$ $$ 3 = \frac{RB}{2} $$ $$ RB = 3 \times 2 = 6 $$ Now we can find the length of side RT: $$ RT = RB + BT = 6 + 2 = 8 $$
4. Set up the Proportion using the Given Similarity (\( \Delta RST \sim \Delta RBA \)): Using the ratio involving the sides we need (ST and BA) and sides we know (RS and RB, or RT and RA): $$ \frac{ST}{BA} = \frac{RS}{RB} $$ Substitute the known values: $$ \frac{x}{9} = \frac{8}{6} $$ Simplify the ratio \( \frac{8}{6} \): $$ \frac{8}{6} = \frac{4}{3} $$ So the equation becomes: $$ \frac{x}{9} = \frac{4}{3} $$
5. Solve for x: Cross-multiply to solve for \( x \): $$ 3 \times x = 9 \times 4 $$ $$ 3x = 36 $$ $$ x = \frac{36}{3} $$ $$ x = 12 $$ Alternatively, using \( \frac{ST}{BA} = \frac{RT}{RA} \): $$ \frac{x}{9} = \frac{8}{6} $$ $$ \frac{x}{9} = \frac{4}{3} $$ $$ 3x = 36 $$ $$ x = 12 $$
6. Conclusion: The value of \( x \) is 12. This corresponds to option (A).
