Question

1. cameron calculated the distance between point c and point d below.

is the distance correct? if not, what is her mistake?
$d = \sqrt{(-2 + 2)^2+(6 + 3)^2}$
$=\sqrt{(0)^2+(9)^2}$
$=9$

Explanation:

Step1: Recall the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Identify the coordinates of points \(C\) and \(D\)

Assume \(C(x_1,y_1)\) and \(D(x_2,y_2)\). Let's say \(C(- 2,6)\) and \(D(2,-3)\). Then \(x_1=-2,y_1 = 6,x_2=2,y_2=-3\).

Step3: Apply the distance formula correctly

\(d=\sqrt{(2-(-2))^2+(-3 - 6)^2}=\sqrt{(2 + 2)^2+(-9)^2}=\sqrt{4^2+(-9)^2}=\sqrt{16 + 81}=\sqrt{97}\).
Cameron's mistake was in the \(x\)-coordinate subtraction. She used \((-2)+2\) instead of \(2-(-2)\) (or vice - versa depending on the order of subtraction in the distance formula. The formula is \((x_2 - x_1)\) or \((x_1 - x_2)\) (since \((a - b)^2=(b - a)^2\)), but she did an addition for the \(x\)-coordinate part.

Answer:

The distance is not correct. Cameron's mistake was in the \(x\)-coordinate subtraction. She used addition instead of subtraction for the \(x\)-coordinate part in the distance formula. The correct distance \(d=\sqrt{97}\).