Plane perpendiculare intr-un paralelipiped
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Marius Mainea
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Plane perpendiculare intr-un paralelipiped
In paralelipipedul dreptunghic ABCDA'B'C'D' avem :AB=a,BC=b,AA'=c. Notam cu E si F proiectiile punctului D pe AC , respectiv pe A'C si cu P si Q proiectiile punctului C'pe B'D', respectiv pe BD'. Aratati ca planele (DEF) si (C'PQ) sunt perpendiculare daca si numai daca \( b^2=a^2+c^2. \)
\( DE\perp AC \) si \( DE\perp AA\prime \Rightarrow DE \perp (AA\prime C) \Rightarrow DE\perp A\prime C \) si \( DF\perp A\prime C \Rightarrow A\prime C\perp (DEF) \)
\( C\prime P\perp D\prime B\prime \) si \( C\prime P\perp BB\prime \Rightarrow C\prime P\prime \perp (D\prime BB\prime) \Rightarrow C\prime P\perp D\prime B \) si \( D\prime B \perp C\prime Q \Rightarrow D\prime B\perp (C\prime PQ) \)
\( (DEF)\perp (C\prime PQ) \Leftrightarrow D\prime B\perp A\prime C \)
Luand \( O \) intersectia diagonalelor \( \Rightarrow OB=\frac{\sqrt{a^2+b^2+c^2}}{2} \Rightarrow \) In \( \Delta COB \Rightarrow a^2+b^2+c^2=2b^2 \Rightarrow a^2+c^2=b^2 \)
\( C\prime P\perp D\prime B\prime \) si \( C\prime P\perp BB\prime \Rightarrow C\prime P\prime \perp (D\prime BB\prime) \Rightarrow C\prime P\perp D\prime B \) si \( D\prime B \perp C\prime Q \Rightarrow D\prime B\perp (C\prime PQ) \)
\( (DEF)\perp (C\prime PQ) \Leftrightarrow D\prime B\perp A\prime C \)
Luand \( O \) intersectia diagonalelor \( \Rightarrow OB=\frac{\sqrt{a^2+b^2+c^2}}{2} \Rightarrow \) In \( \Delta COB \Rightarrow a^2+b^2+c^2=2b^2 \Rightarrow a^2+c^2=b^2 \)
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