The distance formula for the distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a plane is given as:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Let's analyze each given statement to determine its accuracy:
1. It is not a precise definition because it uses variables to represent unknown values.
- This statement is not accurate. Using variables to represent unknown values is a standard practice in mathematics to formulate general rules and equations. The distance formula is mathematically precise even though it uses variables since it provides a specific method to compute the distance based on given coordinates.
2. It is not a precise definition because it uses a square root sign, which means the result might be an irrational number.
- This statement is also not accurate. The use of a square root sign does not affect the precision of the definition. The distance formula provides an exact value for the distance between two points. While the result might indeed be an irrational number, it is still a precise and accurate value.
3. It is not a precise definition because it is based on the difference of two coordinates.
- This statement is not correct. The computation of the difference between two coordinates is a fundamental part of finding the distance between them. The difference allows us to measure the relative position of the points along each axis, and it's essential for the precise calculation of distance.
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.
- While the distance formula uses coordinates, this does not make it imprecise. Coordinates themselves are well-defined mathematical constructs, and the distance formula precisely calculates distances by leveraging these constructs.
Given these analyses, none of the provided statements truthfully critiques the precision of the distance formula. The formula is mathematically precise and robust, regardless of its use of variables, square roots, or coordinate differences.
