Question

1. what is the distance between the two points graphed? what is the location of the point halfway between them?

Explanation:

1. First, assume the coordinates of the two - points:

• Let's assume the grid has a scale of 1 unit per grid square. Suppose the first point has coordinates \((x_1,y_1)\) and the second point has coordinates \((x_2,y_2)\). Counting the grid - squares, if the first point is at \((- 3,2)\) and the second point is at \((5,-2)\).

1. Calculate the distance between the two points using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\):

• Substitute \(x_1=-3,y_1 = 2,x_2 = 5,y_2=-2\) into the formula.
• First, calculate \((x_2 - x_1)\) and \((y_2 - y_1)\): \(x_2 - x_1=5-(-3)=8\) and \(y_2 - y_1=-2 - 2=-4\).
• Then, \((x_2 - x_1)^2+(y_2 - y_1)^2=8^2+(-4)^2=64 + 16=80\).
• So, \(d=\sqrt{80}=4\sqrt{5}\approx8.94\).

1. Calculate the mid - point (the point halfway between them) using the mid - point formula \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\):

• Substitute \(x_1=-3,y_1 = 2,x_2 = 5,y_2=-2\) into the formula.
• \(\frac{x_1 + x_2}{2}=\frac{-3 + 5}{2}=1\) and \(\frac{y_1 + y_2}{2}=\frac{2+( - 2)}{2}=0\).
• The mid - point is \((1,0)\).

Step1: Determine coordinates

Assume first point \((-3,2)\) and second point \((5,-2)\)

Step2: Calculate distance

Use \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), substitute values: \(d=\sqrt{(5 + 3)^2+(-2 - 2)^2}=\sqrt{64 + 16}=\sqrt{80}=4\sqrt{5}\)

Step3: Calculate mid - point

Use \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\), substitute values: \((\frac{-3 + 5}{2},\frac{2-2}{2})=(1,0)\)

Answer:

The distance between the two points is \(4\sqrt{5}\approx8.94\), and the location of the point halfway between them is \((1,0)\)