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Chapter 32: Chapter 32

Problem 1030

Next question

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 \(\sqrt{89}\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

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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