Question

error analysis describe the error in finding the distance between a(6, 2) and b(1, −4).
image with incorrect calculation: ( ab = (6 - 1)^2 + 2 - (-4)^2 )
( = 5^2 + 6^2 )
( = 25 + 36 )
( = 61 )
options:

• did not use a ruler
• did not find the midpoint
• did not take the square root
• did not make... (obscured text)

the actual length, ( ab ), rounded to the nearest tenth is

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, they computed \((6 - 1)^2+[2-(-4)]^2\) and got \(25 + 36=61\), but they forgot to take the square root of the sum of the squares. The other options (not using a ruler, not finding midpoint, sign - related issues) are not relevant here as the distance formula doesn't involve a ruler or midpoint for direct distance calculation, and the sign handling in the squares was correct (since squaring eliminates sign issues). So the error is "Did not take the square root".

Step 1: Recall the distance formula

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

Step 2: Substitute the values into the formula

First, calculate \((x_1 - x_2)^2=(6 - 1)^2=5^2 = 25\) and \((y_1 - y_2)^2=[2-(-4)]^2=(2 + 4)^2=6^2=36\).
Then, find the sum of these squares: \(25+36 = 61\).

Step 3: Take the square root

Now, take the square root of \(61\): \(AB=\sqrt{61}\approx7.8\) (rounded to the nearest tenth).

Answer:

Did not take the square root.

Part 2: Calculating the Actual Length