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Chapter 32: Problem 1030

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

Short Answer

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

Step by step solution

01

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).
02

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: \[d = \sqrt{(7 - 2)^2 + (11 - 3)^2}\]
03

Simplify the expression

Now that the coordinates have been plugged into the formula, we can simplify the expression: \[d = \sqrt{(5)^2 + (8)^2}\]
04

Calculate the sum of the squares

Next, we square the numbers inside the square root and add them together: \[d = \sqrt{25 + 64}\]
05

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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