https://file.aiquickdraw.com/mathbot/file/1737469565817btcyuixz.jpg

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**.

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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.

Identify the segment bisector of JK. Given JM = 7x + 5 and MK = 8x, find the segment bisector and the length of JM.

Solve a geometry problem involving the segment bisector of line segment JK, where JM = 7x + 5 and MK = 8x. Learn how to determine the bisector and calculate the length of JM using algebraic equations and geometry principles. Suitable for students in Grades 6-8.

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