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Find the distance between the points $(0,5)$ and $(3,9)$ . $\newline$ Write your answer as a whole number or a fully simplified radical expression. Do not round. $\newline$ $\_\_\_\_\_\_$ units

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Distance between two points

Full solution

Q. Find the distance between the points

(0,5)

and

(3,9)

.

\newline

Write your answer as a whole number or a fully simplified radical expression. Do not round.

\newline

\_\_\_\_\_\_

units

Introduction: To find the distance between two points in a coordinate plane, we use the distance formula, which is derived from the Pythagorean theorem. The distance formula is: $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ . For the points $(0,5)$ and $(3,9)$ , we have $(x_1, y_1) = (0, 5)$ and $(x_2, y_2) = (3, 9)$ . Let's plug these values into the formula.
Calculate x-coordinate difference: First, calculate the difference in the x-coordinates: $(x_2-x_1)^2 = (3-0)^2$ . This simplifies to $3^2$ , which equals $9$ .
Calculate y-coordinate difference: Next, calculate the difference in the y-coordinates: y_2-y_1)^2 = (9-5)^2\. This simplifies to \$4^2, which equals $16.
Add squares of differences: Now, add the squares of the differences in the $x$ and $y$ coordinates: $9 + 16$ . This equals $25$ .
Find distance: Finally, take the square root of the sum to find the distance: $\sqrt{25}$ . The square root of $25$ is $5$ . Therefore, the distance between the points $(0,5)$ and $(3,9)$ is $5$ units.

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