Point O is on line segment start overline, N, P, end overline 
NP
 . Given N, P, equals, 4, x, plus, 10, commaNP=4x+10, N, O, equals, 4, x, commaNO=4x, and O, P, equals, x, plus, 5, commaOP=x+5, determine the numerical length of start overline, N, P, end overline, . 
NP

We are given the following information:

- \( NP = 4x + 10 \)
- \( NO = 4x \)
- \( OP = x + 5 \)

Since \( O \) lies on the line segment \( NP \), the sum of the lengths of \( NO \) and \( OP \) must equal the length of \( NP \). This gives the equation:

\[
NO + OP = NP
\]

Substituting the given expressions:

\[
4x + (x + 5) = 4x + 10
\]

Now, simplify and solve for \( x \):

\[
4x + x + 5 = 4x + 10
\]

\[
5x + 5 = 4x + 10
\]

Subtract \( 4x \) from both sides:

\[
x + 5 = 10
\]

Subtract 5 from both sides:

\[
x = 5
\]

Now, substitute \( x = 5 \) back into the expression for \( NP \) to find the numerical length of \( NP \):

\[
NP = 4x + 10 = 4(5) + 10 = 20 + 10 = 30
\]

Thus, the length of \( NP \) is \( \boxed{30} \).

Would you like further explanation or more details on any part? Here are 5 related questions:

1. How do you verify the position of a point on a line segment?
2. What other methods can be used to solve similar problems involving line segments?
3. How does the value of \( x \) affect the lengths of \( NO \) and \( OP \)?
4. What are some applications of this type of geometry problem in real-world situations?
5. How does the relationship between segment lengths help in geometric proofs?

**Tip**: When dealing with line segments and points in geometry, always ensure the sum of parts equals the whole segment length for consistent solutions.

Point O is on line segment NP. Given NP = 4x + 10, NO = 4x, and OP = x + 5, determine the numerical length of NP.

Learn how to find the length of line segment NP given the equations NP = 4x + 10, NO = 4x, and OP = x + 5. This geometry problem involves algebraic techniques and the Segment Addition Postulate, suitable for students in Grades 8-10.

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