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 $DP.$
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.
