Pyramid SABC has right triangular base ABC, with ∠ A B C = 90 ∘ . Sides A B = 3 , B C = 3. Lateral lengths are equal and are equal to 2. Find the angle created by lateral length and the base.

Question

Pyramid SABC has right triangular base ABC, with   ∠      A    B    C    =      90    ∘  . Sides   A  B  =      3    ,  B  C  =  3. Lateral lengths are equal and are equal to 2. Find the angle created by lateral length and the base.

dtal50 · Accepted Answer

Step 1Let K be a middle point of AC.Just   ∡  S  B  K  =  ∡  S  A  K  =  arccos  ⁡            3        2    =      30          ∘        ..BK is a median of   Δ  A  B  C and is not perpendicular to AC, otherwise   A  B  =  B  C, which is a contradiction.By the way,   B  K  ⊥  S  K, but we said about it in the first line.Step 2Let SK′ be an altitude of the pyramid.Thus, since   S  A  =  S  B  =  S  C, we obtain: ΔSAK′≅ΔSBK′ and   Δ  S  A      K    ′    ≅  Δ  S  C      K    ′  , which gives   A      K    ′    =  B      K    ′    =  C      K    ′    ,, which says K′ is a center of the circumcircle for   Δ  A  B  C.Thus,       K    ′    ≡  K.

Pyramid SABC has right triangular base ABC, with angle ABC=90^circ. Sides AB=sqrt{3}, BC=3. Lateral lengths are equal and are equal to 2. Find the angle created by lateral length and the base.

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