Question

find the distance between two points.

1. a(13,2) and b(7,10)
2. c(-6,5) and d(-3,1)
3. e(3,7) and f(6,5)
4. g(-5,4) and h(2,6)
5. i(-8,0) and k(1,4)
6. r(2,3) and s(4,-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 points A(13,2) and B(7,10)

Let $(x_1,y_1)=(13,2)$ and $(x_2,y_2)=(7,10)$. Then $d=\sqrt{(7 - 13)^2+(10 - 2)^2}=\sqrt{(-6)^2+8^2}=\sqrt{36 + 64}=\sqrt{100}=10$.

Step3: Solve for points C(-6,5) and D(-3,1)

Let $(x_1,y_1)=(-6,5)$ and $(x_2,y_2)=(-3,1)$. Then $d=\sqrt{(-3+6)^2+(1 - 5)^2}=\sqrt{3^2+(-4)^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step4: Solve for points E(3,7) and F(6,5)

Let $(x_1,y_1)=(3,7)$ and $(x_2,y_2)=(6,5)$. Then $d=\sqrt{(6 - 3)^2+(5 - 7)^2}=\sqrt{3^2+(-2)^2}=\sqrt{9+4}=\sqrt{13}\approx 3.61$.

Step5: Solve for points G(-5,4) and H(2,6)

Let $(x_1,y_1)=(-5,4)$ and $(x_2,y_2)=(2,6)$. Then $d=\sqrt{(2 + 5)^2+(6 - 4)^2}=\sqrt{7^2+2^2}=\sqrt{49+4}=\sqrt{53}\approx 7.28$.

Step6: Solve for points I(-8,0) and K(1,4)

Let $(x_1,y_1)=(-8,0)$ and $(x_2,y_2)=(1,4)$. Then $d=\sqrt{(1 + 8)^2+(4 - 0)^2}=\sqrt{9^2+4^2}=\sqrt{81 + 16}=\sqrt{97}\approx 9.85$.

Step7: Solve for points R(2,3) and S(4,-1)

Let $(x_1,y_1)=(2,3)$ and $(x_2,y_2)=(4,-1)$. Then $d=\sqrt{(4 - 2)^2+(-1 - 3)^2}=\sqrt{2^2+(-4)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\approx 4.47$.

Answer:

1. $10$
2. $5$
3. $\sqrt{13}\approx 3.61$
4. $\sqrt{53}\approx 7.28$
5. $\sqrt{97}\approx 9.85$
6. $2\sqrt{5}\approx 4.47$