Question

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

the actual length, ab , to the nearest tenth is blank units.

Explanation:

Part 1: Identifying the Error

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). In the given calculation, for the \(x\)-coordinates, they used \(6 - 2\) (which is \(x_1 - y_1\) incorrectly) instead of \(x_2 - x_1=1 - 6\) (or \(6 - 1\), since squaring removes the sign difference). The error is that they did not find the differences of the \(x\)-values (and also the \(y\)-values were miscalculated in terms of the correct difference, but the main issue with \(x\)-coordinates: the difference should be between the two \(x\)-values of the points, not \(x_1 - y_1\)). Wait, re - examining: the two points are \(A(6,2)\) and \(B(1, - 4)\). So \(x_1 = 6,y_1 = 2,x_2 = 1,y_2=-4\). The correct \(x\)-difference is \(x_2 - x_1=1 - 6=-5\) (or \(x_1 - x_2 = 6 - 1 = 5\)), and the correct \(y\)-difference is \(y_2 - y_1=-4 - 2=-6\) (or \(y_1 - y_2=2-(-4) = 6\)). In the given calculation, for the \(x\)-term, they used \(6 - 2\) (which is not the difference of \(x\)-coordinates) and for the \(y\)-term, they used \(1-(-4)\) (which is not the difference of \(y\)-coordinates). So the error is "Did not find differences of the \(x\)-values and of the \(y\)-values".

Step 1: Recall the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For points \(A(6,2)\) and \(B(1,-4)\), we have \(x_1 = 6,y_1 = 2,x_2 = 1,y_2=-4\).

Step 2: Calculate the differences

First, calculate the difference in \(x\)-coordinates: \(x_2 - x_1=1 - 6=-5\) (or \(x_1 - x_2 = 6 - 1 = 5\), since we will square it). Then, calculate the difference in \(y\)-coordinates: \(y_2 - y_1=-4 - 2=-6\) (or \(y_1 - y_2=2-(-4)=6\), since we will square it).

Step 3: Square the differences

Square the \(x\)-difference: \((x_2 - x_1)^2=(1 - 6)^2=(-5)^2 = 25\) (or \((6 - 1)^2=5^2 = 25\)). Square the \(y\)-difference: \((y_2 - y_1)^2=(-4 - 2)^2=(-6)^2 = 36\) (or \((2-(-4))^2=6^2 = 36\)).

Step 4: Sum the squares and take the square root

Sum the squares: \(25 + 36=61\). Then take the square root: \(d=\sqrt{61}\approx7.8\) (since \(\sqrt{61}\approx7.81\), and to the nearest tenth is \(7.8\)).

Answer:

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

Part 2: Calculating the Actual Length