Question

use the distance formula to find the distance between each pair of points.
1 a(2, 3) and b(5, 7)
2 v(-2, -6) and w(6, 9)

1. j(-3, 2) and k(6, 5)

4 r(-7, -3) and s(-1, -3)
find the coordinates of the midpoint of a segment having the given endpoints.

1. a(2, 3) and b(5, 7)
2. v(-2, -6) and w(6, 9)
3. j(-3, 2) and k(6, 5)
4. r(-7, -3) and s(-1, -3)

Explanation:

Step1: Recall 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: Calculate distance for A(2,3) and B(5,7)

Let $(x_1,y_1)=(2,3)$ and $(x_2,y_2)=(5,7)$. Then $d=\sqrt{(5 - 2)^2+(7 - 3)^2}=\sqrt{3^2+4^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step3: Calculate distance for V(-2,-6) and W(6,9)

Let $(x_1,y_1)=(-2,-6)$ and $(x_2,y_2)=(6,9)$. Then $d=\sqrt{(6-(-2))^2+(9 - (-6))^2}=\sqrt{(6 + 2)^2+(9+6)^2}=\sqrt{8^2+15^2}=\sqrt{64 + 225}=\sqrt{289}=17$.

Step4: Calculate distance for J(-3,2) and K(6,5)

Let $(x_1,y_1)=(-3,2)$ and $(x_2,y_2)=(6,5)$. Then $d=\sqrt{(6-(-3))^2+(5 - 2)^2}=\sqrt{(6 + 3)^2+3^2}=\sqrt{9^2+3^2}=\sqrt{81+9}=\sqrt{90}=3\sqrt{10}$.

Step5: Calculate distance for R(-7,-3) and S(-1,-3)

Let $(x_1,y_1)=(-7,-3)$ and $(x_2,y_2)=(-1,-3)$. Then $d=\sqrt{(-1-(-7))^2+(-3-(-3))^2}=\sqrt{(-1 + 7)^2+0^2}=\sqrt{6^2}=6$.

Step6: Recall mid - point formula

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Step7: Calculate mid - point for A(2,3) and B(5,7)

$M=(\frac{2 + 5}{2},\frac{3+7}{2})=(\frac{7}{2},5)$.

Step8: Calculate mid - point for V(-2,-6) and W(6,9)

$M=(\frac{-2 + 6}{2},\frac{-6+9}{2})=(2,\frac{3}{2})$.

Step9: Calculate mid - point for J(-3,2) and K(6,5)

$M=(\frac{-3 + 6}{2},\frac{2+5}{2})=(\frac{3}{2},\frac{7}{2})$.

Step10: Calculate mid - point for R(-7,-3) and S(-1,-3)

$M=(\frac{-7-1}{2},\frac{-3-3}{2})=(-4,-3)$.

Answer:

1. Distance between A(2,3) and B(5,7) is 5.
2. Distance between V(-2,-6) and W(6,9) is 17.
3. Distance between J(-3,2) and K(6,5) is $3\sqrt{10}$.
4. Distance between R(-7,-3) and S(-1,-3) is 6.
5. Mid - point of segment with endpoints A(2,3) and B(5,7) is $(\frac{7}{2},5)$.
6. Mid - point of segment with endpoints V(-2,-6) and W(6,9) is $(2,\frac{3}{2})$.
7. Mid - point of segment with endpoints J(-3,2) and K(6,5) is $(\frac{3}{2},\frac{7}{2})$.
8. Mid - point of segment with endpoints R(-7,-3) and S(-1,-3) is $(-4,-3)$.