Question

1. $(-1, 2), (2, -4)$

$(3)^2+(6)^2=sqrt{45}$

1. $(0, 4), (2, 3)$
2. $(12, 12), (-3, 1)$
3. $(5, -10), (-5, 4)$
4. $(4, 3), (-3, 4)$
5. $(4, 0), (-4, 1)$
6. $(1, -9), (6, -6)$
7. $(5, 5), (-6, -4)$

Explanation:

The goal is to find the distance between each pair of points using the distance formula: for points $(x_1,y_1)$ and $(x_2,y_2)$, distance $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

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Problem 5: $(-1,2),(2,-4)$

Step1: Identify coordinates

$x_1=-1,y_1=2;x_2=2,y_2=-4$

Step2: Substitute into distance formula

$d=\sqrt{(2-(-1))^2+(-4-2)^2}=\sqrt{3^2+(-6)^2}$

Step3: Calculate values inside root

$\sqrt{9+36}=\sqrt{45}=3\sqrt{5}$

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Problem 6: $(4,3),(-3,4)$

Step1: Identify coordinates

$x_1=4,y_1=3;x_2=-3,y_2=4$

Step2: Substitute into distance formula

$d=\sqrt{(-3-4)^2+(4-3)^2}=\sqrt{(-7)^2+1^2}$

Step3: Calculate values inside root

$\sqrt{49+1}=\sqrt{50}=5\sqrt{2}$

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Problem 7: $(0,4),(2,3)$

Step1: Identify coordinates

$x_1=0,y_1=4;x_2=2,y_2=3$

Step2: Substitute into distance formula

$d=\sqrt{(2-0)^2+(3-4)^2}=\sqrt{2^2+(-1)^2}$

Step3: Calculate values inside root

$\sqrt{4+1}=\sqrt{5}$

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Problem 8: $(4,0),(-4,1)$

Step1: Identify coordinates

$x_1=4,y_1=0;x_2=-4,y_2=1$

Step2: Substitute into distance formula

$d=\sqrt{(-4-4)^2+(1-0)^2}=\sqrt{(-8)^2+1^2}$

Step3: Calculate values inside root

$\sqrt{64+1}=\sqrt{65}$

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Problem 9: $(12,12),(-3,1)$

Step1: Identify coordinates

$x_1=12,y_1=12;x_2=-3,y_2=1$

Step2: Substitute into distance formula

$d=\sqrt{(-3-12)^2+(1-12)^2}=\sqrt{(-15)^2+(-11)^2}$

Step3: Calculate values inside root

$\sqrt{225+121}=\sqrt{346}$

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Problem 10: $(1,-9),(6,-6)$

Step1: Identify coordinates

$x_1=1,y_1=-9;x_2=6,y_2=-6$

Step2: Substitute into distance formula

$d=\sqrt{(6-1)^2+(-6-(-9))^2}=\sqrt{5^2+3^2}$

Step3: Calculate values inside root

$\sqrt{25+9}=\sqrt{34}$

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Problem 11: $(5,-10),(-5,4)$

Step1: Identify coordinates

$x_1=5,y_1=-10;x_2=-5,y_2=4$

Step2: Substitute into distance formula

$d=\sqrt{(-5-5)^2+(4-(-10))^2}=\sqrt{(-10)^2+14^2}$

Step3: Calculate values inside root

$\sqrt{100+196}=\sqrt{296}=2\sqrt{74}$

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Problem 12: $(5,5),(-6,-4)$

Step1: Identify coordinates

$x_1=5,y_1=5;x_2=-6,y_2=-4$

Step2: Substitute into distance formula

$d=\sqrt{(-6-5)^2+(-4-5)^2}=\sqrt{(-11)^2+(-9)^2}$

Step3: Calculate values inside root

$\sqrt{121+81}=\sqrt{202}$

Answer:

1. $3\sqrt{5}$
2. $5\sqrt{2}$
3. $\sqrt{5}$
4. $\sqrt{65}$
5. $\sqrt{346}$
6. $\sqrt{34}$
7. $2\sqrt{74}$
8. $\sqrt{202}$