The correct statement about the distance formula [tex]\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex] is:
It is not a precise definition because it uses variables to represent unknown values.
Here's a detailed, step-by-step explanation:
1. Understanding the Statement: The distance formula is a mathematical expression used to calculate the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a coordinate plane.
2. Components of the Formula:
- [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] are the x-coordinates of the two points.
- [tex]\( y_1 \)[/tex] and [tex]\( y_2 \)[/tex] are the y-coordinates of the two points.
- The expression [tex]\((x_2 - x_1)\)[/tex] represents the horizontal distance between the points.
- The expression [tex]\((y_2 - y_1)\)[/tex] represents the vertical distance between the points.
- Squaring these differences ensures that any negative sign (due to the coordinate differences) does not affect the calculation of the distance.
- Adding these squares gives the sum of the squared distances.
- Taking the square root of this sum yields the Euclidean distance between the two points.
3. Variables Representing Unknown Values:
- The formula uses [tex]\(x_1, y_1, x_2, y_2\)[/tex] which are variables that can take any numerical values depending on the specific points being considered.
- Because these variables represent potentially unknown values (since the formula is general and not tied to specific points), one might argue that the formula does not give a precise numeric value until specific coordinates are provided.
4. Precision and Definition:
- The formula is a general definition and its precision in providing a numerical result depends on the specific values substituted into the variables.
- A definition considered "precise" in this context implies the need for known, specific values for the variables involved.
- Thus, it is argued that the formula is not "precise" because it uses variables that represent unknown or general values, rather than specific, known quantities.
Conclusively, the statement "It is not a precise definition because it uses variables to represent unknown values" accurately captures the essence of why the distance formula might be considered imprecise in a certain context.
