Math Problem Statement

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

Solution

To solve this problem, we need to determine the value of xx. The diagram shows a triangle MNO\triangle MNO with points MM, NN, OO, and PP. From the diagram:

  1. MN=2x+20MN = 2x + 20
  2. NO=xNO = x
  3. OP=4x60OP = 4x - 60

The triangle appears to be a straight line segment, suggesting that MM, NN, OO, and PP are collinear. Hence, MN+NO=OPMN + NO = OP.

Let's solve this equation:

MN+NO=OPMN + NO = OP

Substituting the given expressions:

(2x+20)+x=4x60(2x + 20) + x = 4x - 60

Simplifying this equation:

3x+20=4x603x + 20 = 4x - 60

Subtracting 3x3x from both sides:

20=x6020 = x - 60

Adding 60 to both sides:

x=80x = 80

Thus, the value of xx is 80\boxed{80}.

Reason for the Answer

The reason is based on the property of collinear points on a straight line segment. For points MM, NN, OO, and PP that are collinear, the sum of the lengths of the smaller segments MNMN and NONO must equal the length of the larger segment OPOP.

Would you like further details or have any questions?

Here are five related questions for further practice:

  1. If OP=4x60OP = 4x - 60, what would be the value of OPOP when x=80x = 80?
  2. What is the length of MNMN when x=80x = 80?
  3. How would the equation change if the segment NONO were labeled as NO=2xNO = 2x instead of NO=xNO = 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.