In triangle R S T , angle T is a right angle, point L lies on

The correct answer is $\frac{44}{3}$. It's given that in triangle $RST$, angle $T$ is a right angle. The area of a right triangle can be found using the formula $A=\frac{1}{2}{\mathcal{l}}_1{\mathcal{l}}_2$, where $A$ represents the area of the right triangle, ${\mathcal{l}}_1$ represents the length of one leg of the triangle, and ${\mathcal{l}}_2$ represents the length of the other leg of the triangle. In triangle $RST$, the two legs are $\overline{RT}$ and $\overline{ST}$. Therefore, if the length of $\overline{RT}$ is $72$ and the area of triangle $RST$ is $792$, then $792=\frac{1}{2}\left(72\right)\left(ST\right)$, or $792=\left(36\right)\left(ST\right)$. Dividing both sides of this equation by $36$ yields $22=ST$. Therefore, the length of $\overline{ST}$ is $22$. It's also given that point $L$ lies on $\overline{RS}$, point $K$ lies on $\overline{ST}$, and $\overline{LK}$ is parallel to $\overline{RT}$. It follows that angle $LKS$ is a right angle. Since triangles $RST$ and $LSK$ share angle $S$ and have right angles $T$ and $K$, respectively, triangles $RST$ and $LSK$ are similar triangles. Therefore, the ratio of the length of $\overline{RT}$ to the length of $\overline{LK}$ is equal to the ratio of the length of $\overline{ST}$ to the length of $\overline{SK}$. If the length of $\overline{RT}$ is $72$ and the length of $\overline{LK}$ is $24$, it follows that the ratio of the length of $\overline{RT}$ to the length of $\overline{LK}$ is $\frac{72}{24}$, or $3$, so the ratio of the length of $\overline{ST}$ to the length of $\overline{SK}$ is $3$. Therefore, $\frac{22}{SK}=3$. Multiplying both sides of this equation by $SK$ yields $22=\left(3\right)\left(SK\right)$. Dividing both sides of this equation by $3$ yields $\frac{22}{3}=SK$. Since the length of $\overline{ST}$, $22$, is the sum of the length of $\overline{SK}$, $\frac{22}{3}$, and the length of $\overline{KT}$, it follows that the length of $\overline{KT}$ is $22-\frac{22}{3}$, or $\frac{44}{3}$. Note that 44/3, 14.66, and 14.67 are examples of ways to enter a correct answer.