Solved on Sep 18, 2023

Find the distance between two points $A(2,-3)$ and $B(4,5)$ using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

STEP 1

Assumptions1. The coordinates of point A are (,-3) . The coordinates of point B are (4,5)
3. We are using the Euclidean distance formula to calculate the distance between two points in a plane.

STEP 2

First, we need to find the difference between the x-values of points A and B. This can be done by subtracting the x-coordinate of point A from the x-coordinate of point B.
$x_{2} - x_{1} =4 -2$

STEP 3

Calculate the difference between the x-values.
$x_{2} - x_{1} = -2 =2$

STEP 4

Next, we need to find the difference between the y-values of points A and B. This can be done by subtracting the y-coordinate of point A from the y-coordinate of point B.
$y_{2} - y_{1} = - (-3)$

STEP 5

Calculate the difference between the y-values.
$y_{2} - y_{1} =5 - (-3) =8$

STEP 6

Now that we have the differences between the x-values and y-values, we can substitute these into the distance formula.
$d = \sqrt{(2)^{2} + (8)^{2}}$

STEP 7

Calculate the squares of the differences.
$d = \sqrt{(4) + (64)}$

STEP 8

Add the squares of the differences.
$d = \sqrt{68}$

STEP 9

implify the square root to find the distance, d.
$d =2\sqrt{17}$ The distance, d, of AB is $2\sqrt{17}$ .

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Find the distance between two points $A(2,-3)$ and $B(4,5)$ using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

STEP 1

Assumptions1. The coordinates of point A are (,-3) . The coordinates of point B are (4,5)
3. We are using the Euclidean distance formula to calculate the distance between two points in a plane.

STEP 2

First, we need to find the difference between the x-values of points A and B. This can be done by subtracting the x-coordinate of point A from the x-coordinate of point B.
$x_{2} - x_{1} =4 -2$

STEP 3

Calculate the difference between the x-values.
$x_{2} - x_{1} = -2 =2$

STEP 4

Next, we need to find the difference between the y-values of points A and B. This can be done by subtracting the y-coordinate of point A from the y-coordinate of point B.
$y_{2} - y_{1} = - (-3)$

STEP 5

Calculate the difference between the y-values.
$y_{2} - y_{1} =5 - (-3) =8$

STEP 6

Now that we have the differences between the x-values and y-values, we can substitute these into the distance formula.
$d = \sqrt{(2)^{2} + (8)^{2}}$

STEP 7

Calculate the squares of the differences.
$d = \sqrt{(4) + (64)}$

STEP 8

Add the squares of the differences.
$d = \sqrt{68}$

STEP 9

implify the square root to find the distance, d.
$d =2\sqrt{17}$ The distance, d, of AB is $2\sqrt{17}$ .

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Find the distance between two points A(2,−3)A(2,-3)A(2,−3) and B(4,5)B(4,5)B(4,5) using the distance formula d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2​−x1​)2+(y2​−y1​)2​.

STEP 1

Assumptions1. The coordinates of point A are (,-3) . The coordinates of point B are (4,5)3. We are using the Euclidean distance formula to calculate the distance between two points in a plane.

STEP 2

First, we need to find the difference between the x-values of points A and B. This can be done by subtracting the x-coordinate of point A from the x-coordinate of point B.x2−x1=4−2x_{2} - x_{1} =4 -2x2​−x1​=4−2

STEP 3

Calculate the difference between the x-values.x2−x1=−2=2x_{2} - x_{1} = -2 =2x2​−x1​=−2=2

STEP 4

Next, we need to find the difference between the y-values of points A and B. This can be done by subtracting the y-coordinate of point A from the y-coordinate of point B.y2−y1=−(−3)y_{2} - y_{1} = - (-3)y2​−y1​=−(−3)

STEP 5

Calculate the difference between the y-values.y2−y1=5−(−3)=8y_{2} - y_{1} =5 - (-3) =8y2​−y1​=5−(−3)=8

STEP 6

Now that we have the differences between the x-values and y-values, we can substitute these into the distance formula.d=(2)2+(8)2d = \sqrt{(2)^{2} + (8)^{2}}d=(2)2+(8)2​

STEP 7

Calculate the squares of the differences.d=(4)+(64)d = \sqrt{(4) + (64)}d=(4)+(64)​

STEP 8

Add the squares of the differences.d=68d = \sqrt{68}d=68​

STEP 9

implify the square root to find the distance, d.d=217d =2\sqrt{17}d=217​The distance, d, of AB is 2172\sqrt{17}217​.

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Find the distance between two points $A(2,-3)$ and $B(4,5)$ using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

Assumptions1. The coordinates of point A are (,-3) . The coordinates of point B are (4,5)
3. We are using the Euclidean distance formula to calculate the distance between two points in a plane.

First, we need to find the difference between the x-values of points A and B. This can be done by subtracting the x-coordinate of point A from the x-coordinate of point B.
$x_{2} - x_{1} =4 -2$

Calculate the difference between the x-values.
$x_{2} - x_{1} = -2 =2$

Next, we need to find the difference between the y-values of points A and B. This can be done by subtracting the y-coordinate of point A from the y-coordinate of point B.
$y_{2} - y_{1} = - (-3)$

Calculate the difference between the y-values.
$y_{2} - y_{1} =5 - (-3) =8$

Now that we have the differences between the x-values and y-values, we can substitute these into the distance formula.
$d = \sqrt{(2)^{2} + (8)^{2}}$

Calculate the squares of the differences.
$d = \sqrt{(4) + (64)}$

Add the squares of the differences.
$d = \sqrt{68}$

implify the square root to find the distance, d.
$d =2\sqrt{17}$ The distance, d, of AB is $2\sqrt{17}$ .