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Pyramid Resolution (Posted on 2012-06-14) Difficulty: 3 of 5
In a regular triangular pyramid SABC (S is the vertex), E is the midpoint of the slant height of the face SBC. The points F, L and M respectively lie on the edges AB, AC and SC and, |AC| = 10*|AL|.

It is known that EFLM is an isosceles trapezoid and the length of its base EF is equal to √7.

Find the volume of the pyramid.

No Solution Yet Submitted by K Sengupta    
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Solution Solution | Comment 1 of 2
The volume of SABC is e√2/12 where e is the edge length of SABC.
All we need to do is find the value of e from givens in the problem.

Let PQ denote the vector from point P to the point Q and * the vector dot product. Here are the givens:

(1) SABC is a regular triangular pyramid (regular tetrahedron).

(2) E is the midpoint of the slant height of face SBC (if D is the midpoint of BC, then E is the midpoint of SD).

(3) Points F, L, and M respectively lie on edges AB, AC, and SC.

(4) |AC| = 10*|AL| (not needed).

(5) EFLM is an isosceles trapezoid (all that is needed is that EF and ML are parallel).

(6) |EF| = √7.

From (1),

     If X,Y are in {A,B,C}; then
           SX*SY
= e if X=Y, = e/2 otherwise.

From (2),

     SE = SD = SB + SC.

From (3),

     SF = xSA + (1-x)SB,                                       (8)
     SL = ySA + (1-y)SC, and                                 (9)
     SM = zSC                                                      (10)

             for 0 ≤ x,y,z ≤ 1.

From (7) & (8),

     EF = SF - SE = [xSA + (1-x)SB] - [SB + SC]

            =
xSA + ( - x)SB - SC.                           (11)

From (9) & (10),

     ML = SL - SM = [ySA + (1-y)SC] - zSC

          = ySA + (1 - y - z)SC.                                 (12)

From (5), (11), and (12),

     EF = wML    for some real w

→  xSA + ( - x)SB - SC = w[ySA + (1 - y - z)SC]

→  (x - wy)SA + ( - x)SB - [w(1 - y - z) + ]SC = 0.     (13)

Since the vectors SA, SB, and SC are linearly independent,
all the coefficients in (13) are zero. From (11) we see that
we only need that of SB (therefore x = ). Thus,

     EF = (3SA - SC).
 
From (6),

     7 = |EF| = EF*EF = (3SA - SC)*(3SA - SC)/16

        = (9SA*SA - 6SA*SC + SC*SC)/16

        = e(9 - 3 + 1)/16 = 7e/16

Therefore, e = 4 and

     Volume(SABC) = 4√2/12 = 16√2/3.

QED




Edited on June 15, 2012, 1:36 am

Edited on June 15, 2012, 1:51 am
  Posted by Bractals on 2012-06-15 01:14:35

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