Question

find the missing length indicated.

1. find ln
2. find rq
3. find sr
4. find vw

Explanation:

13)

Step1: Use mid - segment theorem

In a triangle, if a line segment joins the mid - points of two sides of the triangle, then the line segment is parallel to the third side and half its length. Let the triangle be $\triangle LMN$ with mid - segment $EF$. We have the proportion based on the mid - segment property. If $EF$ is the mid - segment, then $LN = 2(x + 2)$. Also, assume some relationship from the figure's equalities. Let's assume the triangle has properties such that we can set up an equation. Since the mid - segment relationship gives us a connection between the sides. If we assume the triangle is divided proportionally by the mid - segment, we know that the side $LN$ and the mid - segment $EF$ are related as follows:
Let's assume the triangle has side lengths related by the mid - segment rule. If we consider the fact that the mid - segment is half of the parallel side. Let the mid - segment be $x+2$ and the parallel side be $LN$.
We know that $LN = 2(x + 2)=2x + 4$. But we also need to find $x$ from other side - length relationships in the triangle. Let's assume we have another relationship from the figure. If we assume that the sides of the triangle satisfy some equalities. For example, if we consider the non - mid - segment sides, we might find that $x+10$ is related to other sides. However, if we just focus on the mid - segment property for the side $LN$ and the mid - segment $EF$:
Since the mid - segment $EF=x + 2$ and $LN$ is the side parallel to $EF$ and twice its length, $LN=2(x + 2)$.
If we assume the triangle is a standard triangle with mid - segment properties, we know that the length of $LN$ is given by the mid - segment formula.

Step2: Simplify the expression

$LN=2x + 4$

Step1: Apply mid - segment theorem

In $\triangle QUS$, if $RQ$ is related to the mid - segment $RS$. Let $RS=x - 2$ and assume $RQ$ is related to it by the mid - segment property. The mid - segment of a triangle is parallel to the third side and half its length. So if $RS$ is a mid - segment, then the side parallel to it (let's say the side that $RQ$ is part of) has a relationship such that if the mid - segment is $x - 2$, the length of the side parallel to it is $2(x - 2)$. But we also know that the side lengths in the triangle are related in a way that we can set up an equation. Let's assume the triangle has equalities based on its construction. If we consider the side lengths $x+3$ and $x - 2$. Since the mid - segment $RS=x - 2$, and the side parallel to it (the side that $RQ$ is part of) is related by the mid - segment rule.
We know that if $RS$ is the mid - segment, then the length of the side parallel to it (which we can use to find $RQ$) is given by the mid - segment formula.

Step2: Simplify the expression

$RQ = 2(x - 2)=2x-4$

Step1: Use mid - segment property

In $\triangle EDF$, if $SR$ is related to the side lengths. Let the side lengths be related by the mid - segment rule. We know that if the mid - segment property holds, and we have side lengths $2x-14$ and $x + 2$. Since the mid - segment of a triangle is parallel to the third side and half its length. Let's assume the side parallel to the segment with length $2x - 14$ and $x + 2$ has a relationship such that we can set up an equation. If we consider the fact that the mid - segment and the side it is parallel to are related proportionally. Let the mid - segment be part of a proportion. If we assume the triangle has equalities based on its construction, we know that the side $SR$ and the other side lengths are related.
We know that if the mid - segment property is applied, we can set up an equation to find $x$ first. Since the mid - segment and the side it is parallel to are related, we have $2x-14=\frac{1}{2}(x + 2)$ (assuming a certain mid - segment relationship). Solving for $x$:
\[

$$\begin{align*} 2x-14&=\frac{1}{2}(x + 2)\\ 4x-28&=x + 2\\ 4x-x&=2 + 28\\ 3x&=30\\ x&=10 \end{align*}$$

\]

Step2: Find the length of $SR$

Substitute $x = 10$ into the expression for $SR$. If $SR=x + 2$, then $SR=10 + 2=12$

Answer:

$LN = 2x+4$

14)