Let's examine each of the statements provided to determine which is true about the distance formula [tex]\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]:
1. "It is not a precise definition because it uses variables to represent unknown values."
- The use of variables [tex]\( x_1, x_2, y_1, \)[/tex] and [tex]\( y_2 \)[/tex] in the distance formula helps in generalizing the formula so that it can apply to any pair of points in a coordinate system. This makes the formula versatile and powerful, rather than imprecise.
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 trying to highlight that the use of the square root in the formula can yield irrational numbers as results. While the formula can indeed result in irrational numbers (like [tex]\(\sqrt{2}\)[/tex]), this does not make the formula imprecise or incorrect. In mathematics, irrational numbers are just as valid as rational numbers, and the presence of such numbers does not affect the precision of the formula itself.
3. "It is not a precise definition because it is based on the difference of two coordinates."
- The distance formula is derived from the Pythagorean theorem and is precisely based on the differences between the coordinates of two points. This is not a shortcoming but rather a foundational aspect of the formula.
4. "It is not a precise definition because it is based on an understanding of coordinates, which are defined by the distance of a line segment."
- While coordinate geometry involves understanding coordinates and distances between points, describing this as an imprecision of the distance formula is inaccurate. The formula itself provides a consistent and precise method for calculating distance.
Examining these statements, the most accurate reason provided is:
"It is not a precise definition because it uses a square root sign, which means the result might be an irrational number."
Although the presence of irrational numbers might make the calculations more complex, it is the most relevant critique in the context of precision as outlined in the question. However, it's important to note that even though this statement is technically true, it does not imply that the formula is invalid or generally imprecise. It simply acknowledges that irrational results can occur, which is a natural aspect of mathematical calculations involving square roots.
