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Calculate the value of x and provide a reason for the answer

To solve this problem, we need to determine the value of \( x \). The diagram shows a triangle \( \triangle MNO \) with points \( M \), \( N \), \( O \), and \( P \). From the diagram:

1. \( MN = 2x + 20 \)
2. \( NO = x \)
3. \( OP = 4x - 60 \)

The triangle appears to be a straight line segment, suggesting that \( M \), \( N \), \( O \), and \( P \) are collinear. Hence, \( MN + NO = OP \).

Let's solve this equation:

\[
MN + NO = OP
\]

Substituting the given expressions:

\[
(2x + 20) + x = 4x - 60
\]

Simplifying this equation:

\[
3x + 20 = 4x - 60
\]

Subtracting \( 3x \) from both sides:

\[
20 = x - 60
\]

Adding 60 to both sides:

\[
x = 80
\]

Thus, the value of \( x \) is \( \boxed{80} \).

### Reason for the Answer
The reason is based on the property of collinear points on a straight line segment. For points \( M \), \( N \), \( O \), and \( P \) that are collinear, the sum of the lengths of the smaller segments \( MN \) and \( NO \) must equal the length of the larger segment \( OP \).

Would you like further details or have any questions?

Here are five related questions for further practice:
1. If \( OP = 4x - 60 \), what would be the value of \( OP \) when \( x = 80 \)?
2. What is the length of \( MN \) when \( x = 80 \)?
3. How would the equation change if the segment \( NO \) were labeled as \( NO = 2x \) instead of \( NO = x \)?
4. What would happen if the points were not collinear? Would the equation still apply?
5. Can you think of a real-world example where you might use the concept of collinear points?

**Tip:** When solving equations involving geometry, always consider the properties of the figures, such as collinearity or parallelism, to determine the correct relationships between the given expressions.

Learn how to solve a triangle geometry problem involving collinear points and linear equations. This problem requires finding the value of x in a triangle where points M, N, O, and P are collinear. Explore step-by-step solution methods and related questions for practice.

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