Question

find the distance between the points. round the answer to two decimal places.

1. (1, 3), (5, 7)
2. (-8, -5), (-4, 16)
3. (10, 6), (1, -4)
4. (3, 2), (8, 2)
5. (9, -3), (-1, 8)
6. (10, 0), (0, 4)
7. (-7, -2), (6, 9)
8. (-6, 5), (8, -3)
9. (-5, -6), (-9, -4)
10. (2, 0), (-7, 1)

Explanation:

Step1: Recall 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 problem 1

For points $(1,3)$ and $(5,7)$, $x_1 = 1,y_1 = 3,x_2 = 5,y_2 = 7$. Then $d=\sqrt{(5 - 1)^2+(7 - 3)^2}=\sqrt{4^2+4^2}=\sqrt{16 + 16}=\sqrt{32}\approx5.66$.

Step3: Solve for problem 2

For points $(-8, - 9)$ and $(-4,16)$, $x_1=-8,y_1 = - 9,x_2=-4,y_2 = 16$. Then $d=\sqrt{(-4+8)^2+(16 + 9)^2}=\sqrt{4^2+25^2}=\sqrt{16+625}=\sqrt{641}\approx25.32$.

Step4: Solve for problem 3

For points $(10,6)$ and $(1,-4)$, $x_1 = 10,y_1 = 6,x_2 = 1,y_2=-4$. Then $d=\sqrt{(1 - 10)^2+(-4 - 6)^2}=\sqrt{(-9)^2+(-10)^2}=\sqrt{81 + 100}=\sqrt{181}\approx13.45$.

Step5: Solve for problem 4

For points $(3,2)$ and $(8,2)$, $x_1 = 3,y_1 = 2,x_2 = 8,y_2 = 2$. Then $d=\sqrt{(8 - 3)^2+(2 - 2)^2}=\sqrt{5^2+0^2}=5$.

Step6: Solve for problem 5

For points $(9,-3)$ and $(-1,8)$, $x_1 = 9,y_1=-3,x_2=-1,y_2 = 8$. Then $d=\sqrt{(-1 - 9)^2+(8 + 3)^2}=\sqrt{(-10)^2+11^2}=\sqrt{100+121}=\sqrt{221}\approx14.87$.

Step7: Solve for problem 6

For points $(10,0)$ and $(0,4)$, $x_1 = 10,y_1 = 0,x_2 = 0,y_2 = 4$. Then $d=\sqrt{(0 - 10)^2+(4 - 0)^2}=\sqrt{(-10)^2+4^2}=\sqrt{100 + 16}=\sqrt{116}\approx10.77$.

Step8: Solve for problem 7

For points $(-7,-2)$ and $(6,9)$, $x_1=-7,y_1=-2,x_2 = 6,y_2 = 9$. Then $d=\sqrt{(6 + 7)^2+(9 + 2)^2}=\sqrt{13^2+11^2}=\sqrt{169+121}=\sqrt{290}\approx17.03$.

Step9: Solve for problem 8

For points $(-6,5)$ and $(8,-3)$, $x_1=-6,y_1 = 5,x_2 = 8,y_2=-3$. Then $d=\sqrt{(8 + 6)^2+(-3 - 5)^2}=\sqrt{14^2+(-8)^2}=\sqrt{196 + 64}=\sqrt{260}\approx16.12$.

Step10: Solve for problem 9

For points $(-5,-6)$ and $(-9,-4)$, $x_1=-5,y_1=-6,x_2=-9,y_2=-4$. Then $d=\sqrt{(-9 + 5)^2+(-4 + 6)^2}=\sqrt{(-4)^2+2^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}\approx4.47$.

Step11: Solve for problem 10

For points $(2,0)$ and $(-7,1)$, $x_1 = 2,y_1 = 0,x_2=-7,y_2 = 1$. Then $d=\sqrt{(-7 - 2)^2+(1 - 0)^2}=\sqrt{(-9)^2+1^2}=\sqrt{81+1}=\sqrt{82}\approx9.06$.

Answer:

1. Approximately $5.66$
2. Approximately $25.32$
3. Approximately $13.45$
4. $5$
5. Approximately $14.87$
6. Approximately $10.77$
7. Approximately $17.03$
8. Approximately $16.12$
9. Approximately $4.47$
10. Approximately $9.06$