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Let $ABCD$ be a rectangle in space, and let $P$ be a point in space. If $AP = 3,$ $BP = 2,$ and $CP = 7,$ then find $DP.$

 Mar 25, 2023
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Let's denote the vertices of the rectangle as A, B, C, and D. Since ABCD is a rectangle, AD and BC are parallel and have the same length. Therefore, we can use the Pythagorean theorem to find the length of AD:

AD² = AP² + DP²

Similarly, we can use the Pythagorean theorem to find the length of BC:

BC² = BP² + CP²

Since AD and BC are equal in length, we have:

AP² + DP² = BP² + CP²

Substituting the given values, we get:

3² + DP² = 2² + 7²

9 + DP² = 4 + 49

DP² = 44

DP = √44

DP = 2√11

Therefore, the length of DP is 2√11.

 Mar 25, 2023

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