Applications in Trigonometry

Tieria

2009-09-06 5:34 am

1. In the figure, VABC is a pyramid with VA = VB = 8 cm, AB = BC = CA = 6 cm and VC = 7 cm. M is the mid-point of AB.
(a) Are VM and CM both perpendicular to AB?
(b) Find the angle between the planes VAM and ABC.
(c) Find the perpendicular distance of the vertex V from the plane ABC.
(d) Find the volume of the pyramid VABC.

http://i617.photobucket.com/albums/tt257/michaelcoco_/1-2.jpg

The answers are (a) yes, (b) 64.6ﾟ, (c) 6.70 cm, (d) 34.8 cm^3.

※ 請列明計算步驟※

回答 (1)

用戶9112

2009-09-06 6:52 am

✔ 最佳答案

(a) Consider triangles VAM and VBM, VA = VB, VM = VM, AM = BM,
VAM and VBM are congruent triangles (SSS)
VM is perpendicular to AB.
Similarly, CAM and CBM are congruent triangles and CM is perpendicular to AB.
(b) Let the angle be x
VM2 = VB2 - BM2
VM2 = 64 - 9 = 55; VM = √55
CM2 = 36 - 9 = 27; CM = √27
cosx = (VM2 + CM2 - VC2) / 2(VM)(VC)
= (55 + 27 - 49) / [2(√55)(√27)]
= 0.4282
x = 64.6
(c) The perpendicular distance of the vertex V from the plane ABC
= VMsin64.6
= √55 x 0.903
= 6.70cm
(d) The volume of the pyramid VABC = (1/3)(height)(base area)
Base area = (1/2)(CM)(AB) = 15.59cm2
Volume = (1/3)(6.70)(15.59) = 34.8cm3

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