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To solve for \( NO \) and \( MN \) in the triangle:

1. **Given Information**:
   - \( MP \) is the perpendicular bisector of \( NO \).
   - \( MP \) divides \( NO \) equally, meaning \( NP = PO \).
   - \( PO = 3 \), so \( NP = 3 \) as well.

2. **Finding \( NO \)**:
   - Since \( NP = PO = 3 \), the total length of \( NO = NP + PO = 3 + 3 = 6 \).

3. **Finding \( MN \)**:
   - In \( \triangle MNP \), where \( MP = 7 \) and \( NP = 3 \), we can apply the Pythagorean theorem to find \( MN \):
     \[
     MN = \sqrt{MP^2 + NP^2} = \sqrt{7^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58}
     \]
   - Therefore, \( MN = \sqrt{58} \).

### Final Answers
- \( NO = 6 \)
- \( MN = \sqrt{58} \)

Would you like more details on the solution or have additional questions?

### Related Questions
1. How would the solution change if \( MP \) were not the perpendicular bisector?
2. What would \( MO \) be if \( NO \) were a different length?
3. How is the Pythagorean theorem applied in different types of triangles?
4. What properties does a perpendicular bisector bring to a triangle?
5. Can we find the area of \( \triangle MNO \) using the information given?

### Tip
In any triangle with a perpendicular bisector, remember that it splits the opposite side into equal parts, which is helpful for applying symmetry in calculations.

Given MP is the perpendicular bisector of NO, find the following measurements. NO = ?, MN = ?

Solve for the lengths of NO and MN in a triangle where MP is the perpendicular bisector of NO. This problem uses the Pythagorean theorem and concepts of perpendicular bisectors, suitable for Grades 8-10.

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