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Problem 1030

# What is the distance between the points $$(2,3)$$ and $$(7,11) ?$$

Expert verified
The distance between the points $$(2,3)$$ and $$(7,11)$$ is $$\sqrt{89}$$.
See the step by step solution

## Step 1: Identify the coordinates of the two points

We are given the coordinates of two points in the Cartesian coordinate system: Point A is $$(2,3)$$, and Point B is $$(7,11)$$. So, we have: - $$x_1 = 2,$$ $$y_1 = 3;$$ (Point A), - $$x_2 = 7,$$ $$y_2 = 11;$$ (Point B).

## Step 2: Plug in the coordinates into the distance formula

Now that we have the coordinates of both points, we can plug them into the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substitute the values of the coordinates in the formula as follows: $d = \sqrt{(7 - 2)^2 + (11 - 3)^2}$

## Step 3: Simplify the expression

Now that the coordinates have been plugged into the formula, we can simplify the expression: $d = \sqrt{(5)^2 + (8)^2}$

## Step 4: Calculate the sum of the squares

Next, we square the numbers inside the square root and add them together: $d = \sqrt{25 + 64}$

## Step 5: Calculate the square root

Finally, we calculate the square root of the sum: $d = \sqrt{89}$ The distance between the points $$(2,3)$$ and $$(7,11)$$ is $$\sqrt{89}$$.

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