Question

find the distance between each pair of points. 1) 2)

Explanation:

1. First, assume the two - point formula for the distance \(d\) 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 the first graph:
• Let the first point be \((x_1,y_1)=(-3,-1)\) and the second point be \((x_2,y_2)=(2,-3)\).
• Step 1: Calculate the difference in \(x\) - coordinates
• Find \(x_2 - x_1\): \(x_2 - x_1=2-(-3)=2 + 3=5\).
• Step 2: Calculate the difference in \(y\) - coordinates
• Find \(y_2 - y_1\): \(y_2 - y_1=-3-(-1)=-3 + 1=-2\).
• Step 3: Apply the distance formula
• \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=\sqrt{5^2+(-2)^2}=\sqrt{25 + 4}=\sqrt{29}\).
• For the second graph:
• Let the first point be \((x_1,y_1)=(2,-3)\) and the second point be \((x_2,y_2)=(5,0)\).
• Step 1: Calculate the difference in \(x\) - coordinates
• Find \(x_2 - x_1\): \(x_2 - x_1=5 - 2=3\).
• Step 2: Calculate the difference in \(y\) - coordinates
• Find \(y_2 - y_1\): \(y_2 - y_1=0-(-3)=0 + 3=3\).
• Step 3: Apply the distance formula
• \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=\sqrt{3^2+3^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\).

Answer:

1. \(\sqrt{29}\)
2. \(3\sqrt{2}\)