Question

error analysis describe the error in finding the distance between a(6, 2) and b(1, −4) .
image with calculation: ab=√(6−2)² + (1−(−4))²
=√4²+5²
=√16+25
=√41
≈6.4 (with red x)
options:

• did not average x -coordinates and y -coordinates to find the midpoint.
• did not find differences of the x -values and of the y -values.
• did not make all signs positive before subtraction.
• did not take the absolute value of the differences.

(some text scratched)

Explanation:

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) (or \(\sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}\) since squaring eliminates sign differences). For points \(A(6,2)\) and \(B(1, - 4)\), the correct differences should be \(x_2 - x_1=1 - 6\) (or \(6 - 1\)) and \(y_2 - y_1=-4 - 2\) (or \(2-(-4)\)). In the given calculation, the \(x\) - differences were calculated as \(6 - 2\) (which is incorrect, it should be a difference of the \(x\) - coordinates of the two points, i.e., \(6-1\) or \(1 - 6\)) and the \(y\) - differences were calculated as \(1-(-4)\) (incorrect, should be \(2-(-4)\) or \(-4 - 2\)). So the error is that they did not find differences of the \(x\) - values and of the \(y\) - values (they used incorrect values for the differences, like \(6 - 2\) instead of \(6 - 1\) for \(x\) - difference). The first option is about midpoint (not relevant for distance), the third is incorrect as we don't make signs positive before subtraction, the fourth is incorrect as squaring already takes care of sign (we don't need absolute value before squaring).

Answer:

Did not find differences of the \(x\) - values and of the \(y\) - values.