The Correct Answer and Explanation is:
The correct answer is 44/3 units.
Explanation:
To find the length of segment KT, we can follow a logical sequence of steps using the properties of the triangles described.
First, we need to determine the length of the leg ST of the large right triangle RST. The area of a right triangle is calculated by the formula: Area = (1/2) * base * height. For triangle RST, the legs RT and ST serve as the base and height. We are given that the area is 792 square units and the length of RT is 72 units. Plugging these values into the formula allows us to solve for ST:
792 = (1/2) * 72 * ST
792 = 36 * ST
By dividing 792 by 36, we find that the length of ST is 22 units.
Next, we establish the relationship between the smaller triangle SKL and the larger triangle SRT. The problem states that the line segment LK is parallel to the line segment RT. This parallelism creates similar triangles. Specifically, triangle SKL is similar to triangle SRT. This is because they share a common angle at vertex S, and because LK is parallel to RT, the corresponding angles SKL and STR are equal. Since angle T (or STR) is a right angle, angle SKL must also be a right angle.
For similar triangles, the ratio of the lengths of their corresponding sides is constant. We can set up a proportion using the sides LK and RT, and SK and ST:
SK / ST = LK / RT
We are given the lengths of LK (24 units) and RT (72 units). We can use these to find the similarity ratio:
Ratio = 24 / 72 = 1/3.
This means that the sides of the smaller triangle SKL are one-third the length of the corresponding sides of the larger triangle SRT. Therefore, SK = (1/3) * ST.
Using the length of ST we found earlier:
SK = (1/3) * 22 = 22/3 units.
Finally, since point K lies on the segment ST, the length of KT is the difference between the total length of ST and the length of SK.
KT = ST – SK
KT = 22 – (22/3)
KT = (66/3) – (22/3) = 44/3 units.
