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Geometry and Trigonometry Difficulty: Hard

In triangle $R S T$ , angle $T$ is a right angle, point $L$ lies on $\bar{R S}$ , point $K$ lies on $\bar{S T}$ , and $\bar{L K}$ is parallel to $\bar{R T}$ . If the length of $\bar{R T}$ is $72$ units, the length of $\bar{L K}$ is $24$ units, and the area of triangle $R S T$ is $792$ square units, what is the length of $\bar{K T}$ , in units?

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Explanation

The correct answer is $StartFraction 44 Over 3 EndFraction$ . It's given that in triangle $R S T$ , angle $T$ is a right angle. The area of a right triangle can be found using the formula $upper A equals one half script l 1 script l 2$ , where $A$ represents the area of the right triangle, $script l 1$ represents the length of one leg of the triangle, and $script l 2$ represents the length of the other leg of the triangle. In triangle $R S T$ , the two legs are $\bar{R T}$ and $\bar{S T}$ . Therefore, if the length of $\bar{R T}$ is $72$ and the area of triangle $R S T$ is $792$ , then $792 equals one half left parenthesis 72 right parenthesis left parenthesis upper S upper T right parenthesis$ , or $792 equals left parenthesis 36 right parenthesis left parenthesis upper S upper T right parenthesis$ . Dividing both sides of this equation by $36$ yields $22 equals upper S upper T$ . Therefore, the length of $\bar{S T}$ is $22$ . It's also given that point $L$ lies on $\bar{R S}$ , point $K$ lies on $\bar{S T}$ , and $\bar{L K}$ is parallel to $\bar{R T}$ . It follows that angle $L K S$ is a right angle. Since triangles $R S T$ and $L S K$ share angle $S$ and have right angles $T$ and $K$ , respectively, triangles $R S T$ and $L S K$ are similar triangles. Therefore, the ratio of the length of $\bar{R T}$ to the length of $\bar{L K}$ is equal to the ratio of the length of $\bar{S T}$ to the length of $\bar{S K}$ . If the length of $\bar{R T}$ is $72$ and the length of $\bar{L K}$ is $24$ , it follows that the ratio of the length of $\bar{R T}$ to the length of $\bar{L K}$ is $StartFraction 72 Over 24 EndFraction$ , or $3$ , so the ratio of the length of $\bar{S T}$ to the length of $\bar{S K}$ is $3$ . Therefore, $\frac{22}{S K} = 3$ . Multiplying both sides of this equation by $S K$ yields $22 equals left parenthesis 3 right parenthesis left parenthesis upper S upper K right parenthesis$ . Dividing both sides of this equation by $3$ yields $\frac{22}{3} = S K$ . Since the length of $\bar{S T}$ , $22$ , is the sum of the length of $\bar{S K}$ , $StartFraction 22 Over 3 EndFraction$ , and the length of $\bar{K T}$ , it follows that the length of $\bar{K T}$ is $22 - \frac{22}{3}$ , or $StartFraction 44 Over 3 EndFraction$ . Note that 44/3, 14.66, and 14.67 are examples of ways to enter a correct answer.