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Worksheet on 3-D Trigonometry
1.
The diagram shows a rectangular solid ABCDEFGH,
H
with AB = 30 cm, BC = 15 cm and CG = 10 cm.
G
Calculate
10cm
(i) the angle between DG and BG,
E
F
0
giving your answer correct to the nearest 0.1
D
C
(ii) the perpedicular distance of C from BD,
15cm
giving your answer correct to three significant figures
A
B
30cm
(iii) the angle between the planes DGB and DCB, giving your answer correct
to the nearest 0.10 .
(iv) Find the angles between the following pairs of lines :
AG and DF
BD and EG
AC and BH
2.
The rectangle ABCD is a horizontal base of a pyramid, and the vertex V of the pyramid is vertically above
the centre of the base. The length of AB is 6 cm, the length of BC is 12 cm and the length of VA is 20 cm.
The points X on VB and Y on BC are such that AX and YX are both perpendicular to VB.
(i)
Show that the length of BX is 0.9 cm, and calculate the length of BY.
(ii)
Calculate, to the nearest degree, the angle between the planes VAB and VBC.
3.
The diagram shows a tetrahedron ABCD, in which the faces ABC and ABD
B
are congruent triangles, right-angled at C and D respectively,
30cm
30cm
with AC = AD = CD = 40 cm and BC = BD = 30 cm.
(i)
Find the perpendicular distance from D to the line BC,
40cm
C
D
giving 3 significant figures in your answer.
(ii)
Show that the obtuse angle between the planes ABC and ABD
is 112 . 9 0 , correct to the nearest 0.10 .
40cm
40cm
(iii)
Find the angle between DC and the plane ABC, correct to the nearest 0.10 .
(iii)
Find the perpendicular distance of D from the plane ABC,
A
giving 3 significant figures in your answer.
4.
F
The diagram shows a roof, with horizontal rectangular base ABCD,
where AB = 10m, BC=5m. The horizontal ridge EF has length 6m,
C
D
and is at height 4m above the base. The faces AFD and BEC are
inclined at equal angles to the horizontal. The faces ABEF and
CDFE are inclined at equal angles to the horizontal.
B
A
(a)
Find the inclination of the face ABEF to the horizontal,
giving your answer to the nearest 0.1o.
(b)
Find the total area of the four sloping faces of the roof, giving 3 significant figures in your answer.
(c)
Find the lengths of AE and EC, giving 3 significant figures in your answer.
(d)
Find angle AEC, giving your answer to the nearest 0.1o.
5.
The base ABCD of a cube lies in a horizontal plane. The edges AE, BF , CG and DH are vertical, and
EFGH is the top of the cube. Each edge of the cube is length 1 unit. Calculate
E
(i)
(ii)
(iii)
the angle which EC makes with the face CDHG,
the length FX, wher X is the foot of the perpendicular from F to EC,
the angle between the planes CEF and CEH.
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