To solve the problem shown in the image:

Step 1: Determine the segment bisector

The segment bisector of $\overline{JK}$ is the point or line that divides $\overline{JK}$ into two equal parts. Here, $M$ is the midpoint of $\overline{JK}$, so it divides the segment into two equal subsegments: $\overline{JM}$ and $\overline{MK}$.

Step 2: Set the lengths of $\overline{JM}$ and $\overline{MK}$ equal

The given lengths are: $\overline{JM} = 7x + 5, \quad \overline{MK} = 8x$ Since $M$ is the midpoint, these two lengths are equal: $7x + 5 = 8x$

Step 3: Solve for $x$

$7x + 5 = 8x$ Subtract $7x$ from both sides: $5 = x$

Step 4: Calculate the length of $\overline{JM}$

Substitute $x = 5$ into the expression for $\overline{JM}$: $\overline{JM} = 7x + 5 = 7(5) + 5 = 35 + 5 = 40$

Final Answers:

1. The segment bisector of $\overline{JK}$ is point $M$.
2. The length of $\overline{JM}$ is 40 units.

Would you like additional clarification or a breakdown of any step?

Here are five related questions to explore:

1. How do you determine if a point is the midpoint of a line segment?
2. What are other ways to find the bisector of a segment geometrically?
3. Can a line segment have more than one bisector?
4. How would the result change if $\overline{MK}$ were expressed differently (e.g., $8x + 3$)?
5. How is the midpoint formula derived from the equality of the subsegments?

Tip: Always verify that midpoint equations reflect equal subsegment lengths when solving for unknowns.