homework: find the distance between each of the following points. leave your final answer in simplest radical form.
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)

Question

Accepted Answer

# 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: Solve for (1) $(1,3),(5,7)$
Substitute $x_1 = 1,y_1 = 3,x_2 = 5,y_2 = 7$ into the formula:
$$
\begin{align*}
d&=\sqrt{(5 - 1)^2+(7 - 3)^2}\
&=\sqrt{4^2+4^2}\
&=\sqrt{16 + 16}\
&=\sqrt{32}\
&=4\sqrt{2}
\end{align*}
$$
## Step3: Solve for (2) $(-8,-9),(-4,-10)$
Substitute $x_1=-8,y_1 = - 9,x_2=-4,y_2=-10$ into the formula:
$$
\begin{align*}
d&=\sqrt{(-4+8)^2+(-10 + 9)^2}\
&=\sqrt{4^2+(-1)^2}\
&=\sqrt{16+1}\
&=\sqrt{17}
\end{align*}
$$
## Step4: Solve for (3) $(10,6),(1,-4)$
Substitute $x_1 = 10,y_1 = 6,x_2 = 1,y_2=-4$ into the formula:
$$
\begin{align*}
d&=\sqrt{(1 - 10)^2+(-4 - 6)^2}\
&=\sqrt{(-9)^2+(-10)^2}\
&=\sqrt{81 + 100}\
&=\sqrt{181}
\end{align*}
$$
## Step5: Solve for (4) $(3,2),(8,2)$
Substitute $x_1 = 3,y_1 = 2,x_2 = 8,y_2 = 2$ into the formula:
$$
\begin{align*}
d&=\sqrt{(8 - 3)^2+(2 - 2)^2}\
&=\sqrt{5^2+0^2}\
&=5
\end{align*}
$$
## Step6: Solve for (5) $(9,-3),(-1,8)$
Substitute $x_1 = 9,y_1=-3,x_2=-1,y_2 = 8$ into the formula:
$$
\begin{align*}
d&=\sqrt{(-1 - 9)^2+(8 + 3)^2}\
&=\sqrt{(-10)^2+11^2}\
&=\sqrt{100+121}\
&=\sqrt{221}
\end{align*}
$$
## Step7: Solve for (6) $(10,0),(0,4)$
Substitute $x_1 = 10,y_1 = 0,x_2 = 0,y_2 = 4$ into the formula:
$$
\begin{align*}
d&=\sqrt{(0 - 10)^2+(4 - 0)^2}\
&=\sqrt{(-10)^2+4^2}\
&=\sqrt{100 + 16}\
&=\sqrt{116}\
&=2\sqrt{29}
\end{align*}
$$
## Step8: Solve for (7) $(-7,-2),(6,9)$
Substitute $x_1=-7,y_1=-2,x_2 = 6,y_2 = 9$ into the formula:
$$
\begin{align*}
d&=\sqrt{(6 + 7)^2+(9 + 2)^2}\
&=\sqrt{13^2+11^2}\
&=\sqrt{169+121}\
&=\sqrt{290}
\end{align*}
$$
## Step9: Solve for (8) $(-6,5),(8,-3)$
Substitute $x_1=-6,y_1 = 5,x_2 = 8,y_2=-3$ into the formula:
$$
\begin{align*}
d&=\sqrt{(8 + 6)^2+(-3 - 5)^2}\
&=\sqrt{14^2+(-8)^2}\
&=\sqrt{196 + 64}\
&=\sqrt{260}\
&=2\sqrt{65}
\end{align*}
$$
## Step10: Solve for (9) $(-5,-6),(-9,-4)$
Substitute $x_1=-5,y_1=-6,x_2=-9,y_2=-4$ into the formula:
$$
\begin{align*}
d&=\sqrt{(-9 + 5)^2+(-4 + 6)^2}\
&=\sqrt{(-4)^2+2^2}\
&=\sqrt{16 + 4}\
&=\sqrt{20}\
&=2\sqrt{5}
\end{align*}
$$
## Step11: Solve for (10) $(2,0),(-7,1)$
Substitute $x_1 = 2,y_1 = 0,x_2=-7,y_2 = 1$ into the formula:
$$
\begin{align*}
d&=\sqrt{(-7 - 2)^2+(1 - 0)^2}\
&=\sqrt{(-9)^2+1^2}\
&=\sqrt{81+1}\
&=\sqrt{82}
\end{align*}
$$
# Answer:
1. $4\sqrt{2}$
2. $\sqrt{17}$
3. $\sqrt{181}$
4. $5$
5. $\sqrt{221}$
6. $2\sqrt{29}$
7. $\sqrt{290}$
8. $2\sqrt{65}$
9. $2\sqrt{5}$
10. $\sqrt{82}$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall distance formula

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

Step3: Solve for (2) $(-8,-9),(-4,-10)$

Step4: Solve for (3) $(10,6),(1,-4)$

Step5: Solve for (4) $(3,2),(8,2)$

Step6: Solve for (5) $(9,-3),(-1,8)$

Step7: Solve for (6) $(10,0),(0,4)$

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

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

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

Step11: Solve for (10) $(2,0),(-7,1)$

Answer: