Question

example: find the distance between the points (5, -1) and (3, 7).
distance = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}
=\sqrt{(3 - 5)^2+(7 + 1)^2}
=\sqrt{(-2)^2+(8)^2}=\sqrt{4 + 64}=\sqrt{68}\approx8.25 units
find the distance between the points. round the answer to two decimal places.

1. (1, 3), (5, 7)
2. (-8, -9), (-4, -10)
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 $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Solve for (1,3) and (5,7)

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

Step3: Solve for (-8,-9) and (-4,-10)

Let $(x_1,y_1)=(-8,-9)$ and $(x_2,y_2)=(-4,-10)$. Then $d=\sqrt{(-4+8)^2+(-10 + 9)^2}=\sqrt{4^2+(-1)^2}=\sqrt{16+1}=\sqrt{17}\approx4.12$.

Step4: Solve for (10,6) and (1,-4)

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

Step5: Solve for (3,2) and (8,2)

Let $(x_1,y_1)=(3,2)$ and $(x_2,y_2)=(8,2)$. Then $d=\sqrt{(8 - 3)^2+(2 - 2)^2}=\sqrt{5^2+0^2}=5.00$.

Step6: Solve for (9,-3) and (-1,8)

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

Step7: Solve for (10,0) and (0,4)

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

Step8: Solve for (-7,-2) and (6,9)

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

Step9: Solve for (-6,5) and (8,-3)

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

Step10: Solve for (-5,-6) and (-9,-4)

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

Step11: Solve for (2,0) and (-7,1)

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

Answer:

1. $5.66$
2. $4.12$
3. $13.45$
4. $5.00$
5. $14.87$
6. $10.77$
7. $17.03$
8. $16.12$
9. $4.47$
10. $9.06$