Question

using the distance formula
find the distance between the points. for decimal points, round to the nearest hundredth.

1. (4,2), (-6,-5)
2. (2,9), (7,6)
3. (5,3), (-9,7)
4. (-8,3), (2,6)
5. (-1,5), (7,-2)
6. (-9,3), (6,7)
7. (2,-6), (3,8)
8. (-4,1), (-6,-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: Substitute the values for problem 1

For the points \((4,2)\) and \((-6,-5)\), \(x_1 = 4,y_1=2,x_2=-6,y_2 = -5\)
\(d=\sqrt{(-6 - 4)^2+(-5 - 2)^2}=\sqrt{(-10)^2+(-7)^2}=\sqrt{100 + 49}=\sqrt{149}\approx12.20\)

Step3: Substitute the values for problem 2

For the points \((2,9)\) and \((7,6)\), \(x_1 = 2,y_1=9,x_2=7,y_2 = 6\)
\(d=\sqrt{(7 - 2)^2+(6 - 9)^2}=\sqrt{5^2+(-3)^2}=\sqrt{25+9}=\sqrt{34}\approx5.83\)

Step4: Substitute the values for problem 3

For the points \((5,3)\) and \((-9,7)\), \(x_1 = 5,y_1=3,x_2=-9,y_2 = 7\)
\(d=\sqrt{(-9 - 5)^2+(7 - 3)^2}=\sqrt{(-14)^2+4^2}=\sqrt{196 + 16}=\sqrt{212}\approx14.56\)

Step5: Substitute the values for problem 4

For the points \((-8,3)\) and \((2,6)\), \(x_1=-8,y_1 = 3,x_2=2,y_2=6\)
\(d=\sqrt{(2+8)^2+(6 - 3)^2}=\sqrt{10^2+3^2}=\sqrt{100+9}=\sqrt{109}\approx10.44\)

Step6: Substitute the values for problem 5

For the points \((-1,5)\) and \((7,-2)\), \(x_1=-1,y_1 = 5,x_2=7,y_2=-2\)
\(d=\sqrt{(7 + 1)^2+(-2 - 5)^2}=\sqrt{8^2+(-7)^2}=\sqrt{64+49}=\sqrt{113}\approx10.63\)

Step7: Substitute the values for problem 6

For the points \((-9,3)\) and \((6,7)\), \(x_1=-9,y_1 = 3,x_2=6,y_2=7\)
\(d=\sqrt{(6 + 9)^2+(7 - 3)^2}=\sqrt{15^2+4^2}=\sqrt{225+16}=\sqrt{241}\approx15.52\)

Step8: Substitute the values for problem 7

For the points \((2,-6)\) and \((3,8)\), \(x_1=2,y_1=-6,x_2=3,y_2 = 8\)
\(d=\sqrt{(3 - 2)^2+(8 + 6)^2}=\sqrt{1^2+14^2}=\sqrt{1+196}=\sqrt{197}\approx14.04\)

Step9: Substitute the values for problem 8

For the points \((-4,1)\) and \((-6,-9)\), \(x_1=-4,y_1 = 1,x_2=-6,y_2=-9\)
\(d=\sqrt{(-6 + 4)^2+(-9 - 1)^2}=\sqrt{(-2)^2+(-10)^2}=\sqrt{4 + 100}=\sqrt{104}\approx10.20\)

Answer:

1. \(\approx12.20\)
2. \(\approx5.83\)
3. \(\approx14.56\)
4. \(\approx10.44\)
5. \(\approx10.63\)
6. \(\approx15.52\)
7. \(\approx14.04\)
8. \(\approx10.20\)