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已知三棱锥

P-A B C

顶点均在一个半径为 5 的球面上，

A B \perp B C, A C=8, P

到底面

A B C

的距离为 5 ，...

Mar 8, 2024

已知三棱锥

P-A B C

顶点均在一个半径为 5 的球面上，

A B \perp B C, A C=8, P

到底面

A B C

的距离为 5 ，则

P A^{2}+P B^{2}+P C^{2}

的最小值为

Solution by Steps

step 1

Given that

PA=PB=PC=5

and

AC=8

, we can find the length of

AB

and

BC

using the Pythagorean theorem since

AB \perp BC

step 2

Since

AB^2 + BC^2 = 25

and

AB \cdot BC = 0

, one of these lengths must be 0, and the other must be 5. Without loss of generality, let

AB=5

and

BC=0

step 3

The base

ABC

is a right triangle with legs

AB=5

BC=0

, and hypotenuse

AC=8

. This is not possible according to the Pythagorean theorem, indicating an error in the given conditions

step 4

Since the conditions are contradictory, we cannot proceed with the calculation. The problem as stated has no solution

Answer

The problem has no solution due to contradictory conditions.

Key Concept

Pythagorean theorem and the properties of a right triangle

Explanation

The given lengths do not satisfy the Pythagorean theorem for a right triangle, indicating an error in the problem's conditions.