Question

error analysis describe the error in finding the distance between a(6, 2) and b(1, −4) .
image with: ( 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 all the signs positive.

the actual length, ( ab ), rounded 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, they computed \((6 - 1)^2+[2-(-4)]^2\) and got \(25 + 36=61\), but they forgot to take the square root of \(61\) to get the actual distance. 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 \(AB=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(A(6,2)\) 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 \((x_2 - x_1)\) and \((y_2 - y_1)\):
\(x_2 - x_1=1 - 6=-5\), \((x_2 - x_1)^2=(-5)^2 = 25\)
\(y_2 - y_1=-4 - 2=-6\), \((y_2 - y_1)^2=(-6)^2 = 36\)

Step 3: Sum the squares and take the square root

Sum of squares: \(25 + 36=61\)
Take the square root: \(AB=\sqrt{61}\approx7.8\) (rounded to the nearest tenth)

Answer:

Did not take the square root

Part 2: Calculating the Actual Length