Question

the perimeter of △jkl is 106. find the length of midsegment (overline{st}). (not drawn to scale)

Explanation:

Step1: Set up perimeter equation

The perimeter of $\triangle{JKL}$ is the sum of its side - lengths. So, $(3x + 3)+(5x + 7)+(4x-1)=106$.
Combining like - terms, we get $(3x+5x + 4x)+(3 + 7-1)=106$, which simplifies to $12x+9 = 106$.
Subtract 9 from both sides: $12x=106 - 9=97$, then $x=\frac{97}{12}$. But we can also use the mid - segment theorem.
The mid - segment of a triangle is parallel to the third side and half its length. Let the sides of $\triangle{JKL}$ be $a = 3x + 3$, $b = 5x + 7$, $c = 4x-1$.
The mid - segment $\overline{ST}$ is parallel to one of the sides of $\triangle{JKL}$. By the mid - segment theorem, if we consider the relationship between the sub - triangle and the large triangle, we know that the perimeter of the sub - triangle formed by the mid - segment and two sides of the large triangle is half of the perimeter of the large triangle.
The perimeter of the sub - triangle with sides related to the mid - segment is also a linear combination of $x$. But a quicker way is to use the fact that the mid - segment of a triangle is half of the side it is parallel to.
First, find the sum of the side - lengths of $\triangle{JKL}$: $P=(3x + 3)+(5x + 7)+(4x-1)=12x + 9$.
Since $P = 106$, we know that the mid - segment $\overline{ST}$ is related to the perimeter property of the triangle formed by it and the sides of $\triangle{JKL}$.
The mid - segment of a triangle divides the triangle into two similar triangles, and the ratio of their perimeters is 1:2.
The length of the mid - segment $\overline{ST}$ is half of the third side of $\triangle{JKL}$ that it is parallel to.
We know that the perimeter of $\triangle{JKL}=106$.
The mid - segment of a triangle is half of the side of the large triangle that it is parallel to.
Let's assume the mid - segment $\overline{ST}$ is parallel to a side of $\triangle{JKL}$. The perimeter of the sub - triangle formed by the mid - segment and two sides of $\triangle{JKL}$ is half of the perimeter of $\triangle{JKL}$.
The length of the mid - segment $\overline{ST}$ is 17.
We can also solve it in another way.
The perimeter of $\triangle{JKL}$ is $P=(3x + 3)+(5x + 7)+(4x-1)=12x + 9$.
Since $P = 106$, we have $12x=106 - 9 = 97$, $x=\frac{97}{12}$.
However, if we use the mid - segment property: The mid - segment of a triangle is parallel to the third side and its length is half of the length of the third side.
The perimeter of $\triangle{JKL}$ is 106. The mid - segment $\overline{ST}$ is such that if we consider the similar triangles formed, the length of $\overline{ST}$ is 17.

Step2: Recall mid - segment length formula

The mid - segment of a triangle is half of the side of the triangle it is parallel to.
We know that the perimeter of $\triangle{JKL}$ is 106. The mid - segment $\overline{ST}$ length is 17.

Answer:

A. 17