Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 2.8 Junior Find the answer

Given two points in space $P\_1(-1, 3, 5)$ and $P\_2(2, 4, -3)$ , find the distance between them, denoted as $|P\_1P\_2|$ . The distance is equal to $(\ \ \ )$ .

Pick one

Solution

First, we can use the distance formula for two points in three-dimensional space:

$|P\_1P\_2| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Plugging in the given coordinates,

$|P\_1P\_2| = \sqrt{(2 - (-1))^2 + (4 - 3)^2 + (-3 - 5)^2}$

Simplifying the expression,

$|P\_1P\_2| = \sqrt{(3)^2 + (1)^2 + (-8)^2}$

$|P\_1P\_2| = \sqrt{9 + 1 + 64}$

$|P\_1P\_2| = \sqrt{74}$

Thus, the distance between $P\_1$ and $P\_2$ is $\boxed{\sqrt{74}}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.