Tak Sun Secondary School
Form 3 Mathematics
Worksheet 10.4
Chapter 10 More about 3-D Figures
Name : ___________________________
1.
Class : _________ No. :_______
The figure shows two non-parallel planes ABEF and ABCD.
(a) RP is the projection of QP on plane ABCD.
(b) AB is the line of intersection of the two planes.
(c) QPR is the angle between the two planes.
2.
The figure shows a cube ABCDHEFG.
(a) Name the projection of point A on plane EFGH.
(b) Name the projection of point F on plane CDHG.
(a) ∵ ABCDHEFG is a cube.
∴ AE is perpendicular to plane EFGH.
∴ Point E is the projection of point A on plane EFGH.
(b) ∵ ABCDHEFG is a cube.
∴ FG is perpendicular to plane CDHG.
∴ Point G is the projection of point F on plane CDHG.
3.
In the figure, PQRSTUVW is a cube. Name the angles between
planes
(a) UQRT and UVWT,
(b) UQRT and VQRW.
(a) ∵ UQRT and UVWT intersect at line UT, QU ⊥UT and UV ⊥UT.
∴ QUV is the angle between planes UQRT and UVWT.
(b) ∵ UQRT and VQRW intersect at QR, UQ ⊥ QR and VQ ⊥ QR (or TR ⊥ QR and WR⊥ QR).
∴ UQV (or TRW) is the angle between planes UQRT and VQRW.
4.
In the figure, ABCDEF is a prism, and its base ABC is a
right-angled triangle.
(a) Name the projection of BC on plane AEDC.
(b) Name the angle between planes ABFE and AEDC.
(a) ∵ BA is perpendicular to plane AEDC.
∴ AC is the projection of BC on plane AEDC.
(b) ∵ ABFE and AEDC intersect at AE,
BA ⊥ AE and AC ⊥ AE (or FE ⊥ AE and DE ⊥ AE).
∴ BAC (or FED) is the angle between planes ABFE and AEDC.
5.
The figure shows a cuboid ABCDHEFG. For each of the
following planes,
(i)
name the projection of AG on the plane,
(ii) name the angle between AG and the plane.
(a) ABCD
(b) ADHE
(c) ABFE
(a) (i) ∵ ABCDHEFG is a cuboid.
∴ GC is perpendicular to plane ABCD.
∴ AC is the projection of AG on plane ABCD.
(ii) GAC is the angle between AG and plane ABCD.
(b) (i) ∵ ABCDHEFG is a cuboid.
∴ GH is perpendicular to plane ADHE.
∴ AH is the projection of AG on plane ADHE.
(ii) GAH is the angle between AG and plane ADHE.
(c) (i) ∵ ABCDHEFG is a cuboid.
∴ GF is perpendicular to plane ABFE.
∴ AF is the projection of AG on plane ABFE.
(ii) GAF is the angle between AG and plane ABFE.
6.
In the figure, ABCDHEFG is a cuboid.
(a) (i)
Name the projection of BD on plane ADHE.
(ii) Name the angle between BD and plane ADHE.
(b) Name the angle between planes ABCD and BCGF.
∵ ABCDHEFG is a cuboid.
∴ BA is perpendicular to plane ADEH.
∴ AD is the projection of BD on plane ADEH.
(ii) BDA is the angle between BD and plane ADEH.
(a) (i)
(b) ∵ ABCD and BCGF intersect at BC, AB ⊥ BC and FB ⊥ BC (or DC ⊥ BC and GC ⊥ BC).
∴ ABF (or DCG) is the angle between
planes ABCD and BCGF.
7.
In the figure, VPQR is a regular tetrahedron of height VO. VS is
the height of △VQR.
(a) (i)
Name the projection of VS on plane PQR.
(ii) Name the angle between VS and plane PQR.
(b) Name the angle between planes VQR and PQR.
(a) (i) ∵
VO is the height of the regular tetrahedron.
∴
VO is perpendicular to plane PQR.
∴
OS is the projection of VS on plane PQR.
(ii) VSO is the angle between VS and plane PQR.
(b)
∵
∴
∵
∴
VS is the height of ᇞVQR.
VS  RQ
Planes VQR and PQR intersect at RQ,
OS  RQ and VS  RQ.
VSO is the angle between planes VQR and PQR.
8.
The figure shows a square pyramid VABCD. O is the point of
intersection of the diagonals of ABCD. VO is the height of the
pyramid. VP is the height of △VAD.
(a) (i)
Name the projection of VP on plane ABCD.
(ii) Name the angle between VP and plane ABCD.
(b) Name the angle between planes VAD and ABCD.
(a) (i)
∵
VO is the height of the pyramid.
∴
VO is perpendicular to plane ABCD.
∴
OP is the projection of VP on plane ABCD.
(ii) VPO is the angle between VP and plane ABCD.
(b)
9.
∵
VAD and ABCD intersect at AD, VP ⊥AD and OP ⊥AD.
∴
VPO is the angle between planes VAD and ABCD.
In the figure, ABCD is a triangular pyramid, where ∠BAD = 50°,
BD ⊥ CD, AD  BD and AD  CD.
(a) (i)
Name the projection of point A on plane DBC.
(ii) Find the angle between AB and plane DBC.
(b) Find the angle between planes ABD and DBC.
(a) (i)
∵
AD is perpendicular to plane DBC.
∴
Point D is the projection of point A on plane DBC.
(ii) ∵
AD is perpendicular to plane DBC.
∴
DB is the projection of AB on plane DBC.
∴
ABD is the angle between AB and plane DBC.
In ᇞABD,
ADB  ABD  DAB  180
90  ABD  50  180
ABD  40
(b)
( sum of △)
∴
The angle between AB and plane DBC is 40°.
∵
ABD and DBC intersect at DB, AD ⊥ DB and CD ⊥ DB.
∴
The angle between planes ABD and DBC is ADC, i.e. 90°.