Answer :
Let's analyze each of the statements given about the distance formula to determine which one is true.
1. It is not a precise definition because it uses variables to represent unknown values.
- This statement suggests that using variables makes the definition imprecise. However, using variables, such as \(x_1, x_2, y_1,\) and \(y_2\), does not make the definition imprecise. Variables are used to generalize the formula so that it can be applied to any pair of points in the coordinate plane. The formula itself is mathematically precise and correct.
2. It is not a precise definition because it uses a square root sign, which means the result might be an irrational number.
- The use of the square root sign is part of the mathematical definition of distance, derived from the Pythagorean theorem. While the result may indeed be an irrational number, this does not make the definition imprecise. The square root is a precise mathematical operation, and the definition is exact regardless of whether the result is rational or irrational.
3. It is not a precise definition because it is based on the difference of two coordinates.
- The calculation of the distance between two points fundamentally involves measuring the differences between their coordinates. This use of differences is a standard and precise method in mathematics. Thus, this statement is incorrect because the difference of coordinates is a necessary and precise part of the distance formula.
4. It is not a precise definition because it is based on an understanding of coordinates, which are defined based on the distance of a line segment.
- The coordinates themselves are precise and well-defined based on the Cartesian coordinate system. The distance formula is derived using these precise definitions. The understanding of coordinates in the context of the distance formula does not introduce imprecision. Instead, it relies on these precise definitions to accurately calculate the distance.
Considering all the statements, none accurately describe why the distance formula might be considered imprecise. Actually, the distance formula is a precise and well-established mathematical tool used to determine the distance between two points in a plane based on their coordinates. Therefore, none of the provided statements accurately describe the distance formula being imprecise for the given reasons.
1. It is not a precise definition because it uses variables to represent unknown values.
- This statement suggests that using variables makes the definition imprecise. However, using variables, such as \(x_1, x_2, y_1,\) and \(y_2\), does not make the definition imprecise. Variables are used to generalize the formula so that it can be applied to any pair of points in the coordinate plane. The formula itself is mathematically precise and correct.
2. It is not a precise definition because it uses a square root sign, which means the result might be an irrational number.
- The use of the square root sign is part of the mathematical definition of distance, derived from the Pythagorean theorem. While the result may indeed be an irrational number, this does not make the definition imprecise. The square root is a precise mathematical operation, and the definition is exact regardless of whether the result is rational or irrational.
3. It is not a precise definition because it is based on the difference of two coordinates.
- The calculation of the distance between two points fundamentally involves measuring the differences between their coordinates. This use of differences is a standard and precise method in mathematics. Thus, this statement is incorrect because the difference of coordinates is a necessary and precise part of the distance formula.
4. It is not a precise definition because it is based on an understanding of coordinates, which are defined based on the distance of a line segment.
- The coordinates themselves are precise and well-defined based on the Cartesian coordinate system. The distance formula is derived using these precise definitions. The understanding of coordinates in the context of the distance formula does not introduce imprecision. Instead, it relies on these precise definitions to accurately calculate the distance.
Considering all the statements, none accurately describe why the distance formula might be considered imprecise. Actually, the distance formula is a precise and well-established mathematical tool used to determine the distance between two points in a plane based on their coordinates. Therefore, none of the provided statements accurately describe the distance formula being imprecise for the given reasons.
