Find the angle in a Tetrahedron

**beq!x** · April 25th 2009, 09:15 AM

The point $P$ is the midpoint of the altitude of the regular Tetrahedron. $P$ is linked with $A , B$ of the base of Tetrahedron with segments $PA$ and $PB$ .
Find the angles between these segments.

**Nacho** · April 25th 2009, 05:04 PM

Find the angle in a Tetrahedron-aaaaaaaaaaaaaaaaaaaa.png

We say tha size of each one triangule is $l$ , now $AC=CF=CB$ hence $ CB = \frac{{l\sqrt 3 }} {3} \Rightarrow DC = \frac{{l\sqrt 6 }} {3} \Rightarrow PC = \frac{{l\sqrt 6 }} {6} $

for other size

$ PC^2 + CB^2 = PB^2 \Rightarrow PB = \frac{{\sqrt 2 l}} {2} $

but: $ \sin \alpha = \frac{{GB}} {{PB}} = \frac{{\frac{l} {2}}} {{\frac{{\sqrt 2 l}} {2}}} = \frac{{\sqrt 2 }} {2} \Rightarrow \alpha = \frac{\pi } {4} \Leftrightarrow \measuredangle APB = \frac{\pi } {2} $