Ask AI: What is the distance between the points (1,–1) and (–5,9)?

Question

What is the distance between the points (1,–1) and (–5,9)?

gpt-4o · Accepted Answer

To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a Cartesian plane, you can use the distance formula:

$$ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Given the points $(1, -1)$ and $(-5, 9)$, we can substitute $x_1 = 1$, $y_1 = -1$, $x_2 = -5$, and $y_2 = 9$ into the formula:

$$ \text{Distance} = \sqrt{((-5) - 1)^2 + (9 - (-1))^2} $$

First, simplify inside the parentheses:

$$ \text{Distance} = \sqrt{(-6)^2 + (10)^2} $$

Next, calculate the squares of these numbers:

$$ \text{Distance} = \sqrt{36 + 100} $$

Finally, add the values and take the square root:

$$ \text{Distance} = \sqrt{136} $$

$$ \text{Distance} = \sqrt{4 \times 34} $$

$$ \text{Distance} = 2\sqrt{34} $$

So, the distance between the points $(1, -1)$ and $(-5, 9)$ is $2\sqrt{34}$ units.

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