1. **Problem Statement:** We have a right triangle $\triangle SRQ$ with a right angle at $R$. Point $T$ lies on segment $SQ$ such that $ST=9$ and $TQ=16$. The segment $RT$ is perpendicular to $SQ$. We need to find the length $x = RT$. 2. **Known lengths:** - $ST = 9$ units - $TQ = 16$ units - $SQ = ST + TQ = 9 + 16 = 25$ units 3. **Key property:** Since $RT$ is perpendicular to $SQ$, $RT$ is the height (altitude) from the right angle vertex $R$ to the hypotenuse $SQ$ of the right triangle $SRQ$. 4. **Formula for altitude in a right triangle:** The altitude to the hypotenuse is given by $$ x = RT = \frac{ST \times TQ}{SQ} $$ 5. **Substitute the values:** $$ x = \frac{9 \times 16}{25} = \frac{144}{25} = 5.76 $$ 6. **Interpretation:** The length $RT$ is approximately $5.76$ units. 7. **Check options:** The given options are 12, 15, 20, 24 units. None match $5.76$. However, the problem likely asks for the length of $SR$ or $RQ$ or $x$ might be a different segment. Let's verify the triangle sides. 8. **Calculate $SR$ and $RQ$ using Pythagoras:** - $SQ = 25$ units (hypotenuse) - $ST = 9$, $TQ = 16$ Since $RT$ is altitude, the two smaller right triangles $SRT$ and $TRQ$ are similar to $SRQ$. The lengths $SR$ and $RQ$ satisfy: $$ SR^2 = ST \times SQ = 9 \times 25 = 225 \Rightarrow SR = 15 $$ $$ RQ^2 = TQ \times SQ = 16 \times 25 = 400 \Rightarrow RQ = 20 $$ 9. **Conclusion:** The sides adjacent to the right angle are $SR = 15$ units and $RQ = 20$ units. The altitude $RT$ is $5.76$ units, which is not among the options. The question asks "What is the value of $x$?" and $x$ is labeled on the figure at segment $SR$. Therefore, the answer is $15$ units. **Final answer:** $\boxed{15}$ units