Question

the distance formula
find the distance between each pair of points.

1. (7, 3), (-1, -4)
2. (6, -7), (3, -5)
3. (5, -8), (-8, 6)
4. (-7, 0), (-2, -4)
5. (-2, 2), (-6, -8)
6. (-3, -1), (-4, 0)
7. (-2, 3), (-1, 7)

Explanation:

Step1: Recall the distance formula

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

Step2: Solve for the first pair \((7, 3)\) and \((-1, -4)\)

Identify \( x_1 = 7,y_1 = 3,x_2=-1,y_2=-4 \).
Substitute into the formula: \( d=\sqrt{(-1 - 7)^2+(-4 - 3)^2}=\sqrt{(-8)^2+(-7)^2}=\sqrt{64 + 49}=\sqrt{113}\approx10.63 \)

Step3: Solve for the third pair \((6, -7)\) and \((3, -5)\)

Identify \( x_1 = 6,y_1=-7,x_2 = 3,y_2=-5 \).
Substitute into the formula: \( d=\sqrt{(3 - 6)^2+(-5-(-7))^2}=\sqrt{(-3)^2+(2)^2}=\sqrt{9 + 4}=\sqrt{13}\approx3.61 \)

Step4: Solve for the fifth pair \((5, -8)\) and \((-8, 6)\)

Identify \( x_1 = 5,y_1=-8,x_2=-8,y_2 = 6 \).
Substitute into the formula: \( d=\sqrt{(-8 - 5)^2+(6-(-8))^2}=\sqrt{(-13)^2+(14)^2}=\sqrt{169+196}=\sqrt{365}\approx19.10 \)

Step5: Solve for the seventh pair \((-7, 0)\) and \((-2, -4)\)

Identify \( x_1=-7,y_1 = 0,x_2=-2,y_2=-4 \).
Substitute into the formula: \( d=\sqrt{(-2-(-7))^2+(-4 - 0)^2}=\sqrt{(5)^2+(-4)^2}=\sqrt{25 + 16}=\sqrt{41}\approx6.40 \)

Step6: Solve for the ninth pair \((-2, 2)\) and \((-6, -8)\)

Identify \( x_1=-2,y_1 = 2,x_2=-6,y_2=-8 \).
Substitute into the formula: \( d=\sqrt{(-6-(-2))^2+(-8 - 2)^2}=\sqrt{(-4)^2+(-10)^2}=\sqrt{16 + 100}=\sqrt{116}\approx10.77 \)

Step7: Solve for the eleventh pair \((-3, -1)\) and \((-4, 0)\)

Identify \( x_1=-3,y_1=-1,x_2=-4,y_2 = 0 \).
Substitute into the formula: \( d=\sqrt{(-4-(-3))^2+(0-(-1))^2}=\sqrt{(-1)^2+(1)^2}=\sqrt{1 + 1}=\sqrt{2}\approx1.41 \)

Step8: Solve for the thirteenth pair \((-2, 3)\) and \((-1, 7)\)

Identify \( x_1=-2,y_1 = 3,x_2=-1,y_2 = 7 \).
Substitute into the formula: \( d=\sqrt{(-1-(-2))^2+(7 - 3)^2}=\sqrt{(1)^2+(4)^2}=\sqrt{1+16}=\sqrt{17}\approx4.12 \)

Answer:

1. \(\sqrt{113}\) (or approximately \(10.63\))
2. \(\sqrt{13}\) (or approximately \(3.61\))
3. \(\sqrt{365}\) (or approximately \(19.10\))
4. \(\sqrt{41}\) (or approximately \(6.40\))
5. \(\sqrt{116}\) (or approximately \(10.77\))
6. \(\sqrt{2}\) (or approximately \(1.41\))
7. \(\sqrt{17}\) (or approximately \(4.12\))