Example
VABCD is a pyramid, with VA= VB=VC=VD=5cm and ABCD a square of side 4cm. Find the angle between VA and ABCD.
Solution
-Drop a perpendicular from V to ABCD. This meets ABCD at x, the centre of the square. So the projection of VA on ABCD is AX . The angle we want is <VAX.
By Pythagoras theorem in ΔABC
EXERCISE 3.3B
1. 1) ABCDEFGH is a cube with a side of 4cm. Find the angle between the line AG and the face ABCD
2. 2) VABCD is a pyramid with a rectangular base ABCD. AB=20M, AD=30M and VA=VB=VC=VD=25M. Find the angle between VA and ABCD
ANGLE BETWEEN TWO PLANES
Example
The diagram show a cube ABCDEFGH of side 10cm. Find the angles between
a) a) ABCD and ABGH F G
b) b) FHA and FHDB
Solution
A) a)The planes meet in the line AB.
-AD is a line in ABCD which is perpendicular to AB.
-AH is a line in ABGH which is perpendicular to AB. So the angle we want is the angle between AD and AH Which is < DAH.
b). The planes meet in the line FH.
Let x be the midpoint of FH, and let p be the midpoint of BD
-Then XA is a line in FHA which is perpendicular to FH.XP is a line FHDB which is perpendicular to FH
-So the angle we want is the angle between XA and XP, which is <AXP.
-XP is the height of the cube: which is 10cm.
-AP is half the diagonal of ABCD ie
EXERCISE 3.3C
1. The diagram show a pyramid VABCD in which ABCD is a square of side 20m, and VA= VB=VC=VD=15m
Find the angles between the planes
a) VAB and ABCD
b) VAB and VCD
2. 2) A prism has length 8cm, and its cross- section is an equilateral triangle of side 5cm. Find angles between the planes.
a) ADEB and BEFC
b) AEF ad BEFC
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