distance formula
l151
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)

Question

Accepted Answer

# Explanation:
## Step1: Identify coordinates
For problem 1: $(x_1,y_1)=(1,3)$, $(x_2,y_2)=(5,7)$
## Step2: Substitute into distance formula
$$
\begin{align*}
d&=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\
&=\sqrt{(5 - 1)^2+(7 - 3)^2}\
&=\sqrt{4^2+4^2}\
&=\sqrt{16 + 16}\
&=\sqrt{32}\
&\approx5.66
\end{align*}
$$
## Step3: For problem 2: $(x_1,y_1)=(-8,-9)$, $(x_2,y_2)=(-4,-10)$
$$
\begin{align*}
d&=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\
&=\sqrt{(-4+8)^2+(-10 + 9)^2}\
&=\sqrt{4^2+(-1)^2}\
&=\sqrt{16+1}\
&=\sqrt{17}\
&\approx4.12
\end{align*}
$$
## Step4: For problem 3: $(x_1,y_1)=(10,6)$, $(x_2,y_2)=(1,-4)$
$$
\begin{align*}
d&=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\
&=\sqrt{(1 - 10)^2+(-4 - 6)^2}\
&=\sqrt{(-9)^2+(-10)^2}\
&=\sqrt{81+100}\
&=\sqrt{181}\
&\approx13.45
\end{align*}
$$
## Step5: For problem 4: $(x_1,y_1)=(3,2)$, $(x_2,y_2)=(8,2)$
$$
\begin{align*}
d&=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\
&=\sqrt{(8 - 3)^2+(2 - 2)^2}\
&=\sqrt{5^2+0^2}\
&=\sqrt{25}\
&=5.00
\end{align*}
$$
## Step6: For problem 5: $(x_1,y_1)=(9,-3)$, $(x_2,y_2)=(-1,8)$
$$
\begin{align*}
d&=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\
&=\sqrt{(-1 - 9)^2+(8 + 3)^2}\
&=\sqrt{(-10)^2+11^2}\
&=\sqrt{100+121}\
&=\sqrt{221}\
&\approx14.87
\end{align*}
$$
## Step7: For problem 6: $(x_1,y_1)=(10,0)$, $(x_2,y_2)=(0,4)$
$$
\begin{align*}
d&=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\
&=\sqrt{(0 - 10)^2+(4 - 0)^2}\
&=\sqrt{(-10)^2+4^2}\
&=\sqrt{100+16}\
&=\sqrt{116}\
&\approx10.77
\end{align*}
$$
## Step8: For problem 7: $(x_1,y_1)=(-7,-2)$, $(x_2,y_2)=(6,9)$
$$
\begin{align*}
d&=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\
&=\sqrt{(6 + 7)^2+(9 + 2)^2}\
&=\sqrt{13^2+11^2}\
&=\sqrt{169+121}\
&=\sqrt{290}\
&\approx17.03
\end{align*}
$$
## Step9: For problem 8: $(x_1,y_1)=(-6,5)$, $(x_2,y_2)=(8,-3)$
$$
\begin{align*}
d&=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\
&=\sqrt{(8 + 6)^2+(-3 - 5)^2}\
&=\sqrt{14^2+(-8)^2}\
&=\sqrt{196+64}\
&=\sqrt{260}\
&\approx16.12
\end{align*}
$$
## Step10: For problem 9: $(x_1,y_1)=(-5,-6)$, $(x_2,y_2)=(-9,-4)$
$$
\begin{align*}
d&=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\
&=\sqrt{(-9 + 5)^2+(-4 + 6)^2}\
&=\sqrt{(-4)^2+2^2}\
&=\sqrt{16+4}\
&=\sqrt{20}\
&\approx4.47
\end{align*}
$$
## Step11: For problem 10: $(x_1,y_1)=(2,0)$, $(x_2,y_2)=(-7,1)$
$$
\begin{align*}
d&=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\
&=\sqrt{(-7 - 2)^2+(1 - 0)^2}\
&=\sqrt{(-9)^2+1^2}\
&=\sqrt{81+1}\
&=\sqrt{82}\
&\approx9.06
\end{align*}
$$

# 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) $9.06$

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QUESTION IMAGE

Question

Explanation:

Step1: Identify coordinates

Step2: Substitute into distance formula

Step3: For problem 2: \((x_1,y_1)=(-8,-9)\), \((x_2,y_2)=(-4,-10)\)

Step4: For problem 3: \((x_1,y_1)=(10,6)\), \((x_2,y_2)=(1,-4)\)

Step5: For problem 4: \((x_1,y_1)=(3,2)\), \((x_2,y_2)=(8,2)\)

Step6: For problem 5: \((x_1,y_1)=(9,-3)\), \((x_2,y_2)=(-1,8)\)

Step7: For problem 6: \((x_1,y_1)=(10,0)\), \((x_2,y_2)=(0,4)\)

Step8: For problem 7: \((x_1,y_1)=(-7,-2)\), \((x_2,y_2)=(6,9)\)

Step9: For problem 8: \((x_1,y_1)=(-6,5)\), \((x_2,y_2)=(8,-3)\)

Step10: For problem 9: \((x_1,y_1)=(-5,-6)\), \((x_2,y_2)=(-9,-4)\)

Step11: For problem 10: \((x_1,y_1)=(2,0)\), \((x_2,y_2)=(-7,1)\)

Answer: