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PQ applied to T 3
give ( )( ) = ( ) and this is a reflection in the line y = – x.
The inverse of a reflection in the line y = x is itself a reflection in the line y = x.
7 (a) Reflection in the line x + y = 0, i.e. y = – x
(Hint: if in doubt, join corresponding points on each triangle and bisect these lines.)
(b) A = (
3 4 3
) B = ( – 2 2 5
2 2 5
)
– 3 – 4 – 3
a b 3 4 3
( )( ) = ( – 2 2 5
c d
)
2 2 5 – 3 – 4 – 3
3a + 2b 4a + 2b 3a + 5b
( ) = ( – 2 2 5
)
3c + 2d 4c + 2d 3c + 5d – 3 – 4 – 3
Comparing elements:
3a + 2b = 2
4a + 2b = 2
so a = 0 and b = 1
Also 3c + 2d = – 3
4c + 2d = – 4
so c = – 1 and d = 0
Hence the transformation matrix = (
0 1
)
– 1 0
Model answers
(c) By comparing corresponding lengths, the scale factor is – 2. By joining corresponding points with straight lines
and finding the intersection of these lines, the centre is at (1, 5).
2 1 3 4 3
(d) (i) ( 0 3)( 2 2 5) = ( 6 6 15)
1 1
(ii) Area = (10 – 8) × – (15 – 6) = –
2
2 × 2 × 9 = 9 square units
(iii) (
a b
)(
8 10 11
) = (
3 4 3
c d 6 6 15 2 2 5 )
8a + 6b 10a + 6b 11a + 15b
( ) = 3 4 3
( )
8c + 6d 10c + 6d 11c + 15d 2 2 5
Comparing elements:
8a + 6b = 3
10a + 6b = 4
1
so 2a = 1, thus a = –
2
3 – 4
and b = =
– 1
–––––– ––6
6
Also 8c + 6d = 2
10c + 6d = 2
1
so c = 0 = d = –
3
8 (a) (i) A shear along the x-axis direction, shear factor = – 2 (or a shear with the x-axis as the invariant line)
(ii) A shear along the y-axis direction, shear factor = 2 (or a shear with the y-axis as the invariant line)
(b) (
a b
)(
0 3 5 2
) = (
0 3 3 0
c d 0 0 1 1 0 0 2 2 )
3a = 3 thus a = 1
5a + b = 3 thus b = – 2
Also
5c + d = 2 thus c = 0
2c + d = 2 thus d = 2
so the matrix is (
1
– 2
0 2 )
9 (a) (i) A parallelogram
0
– 1
– 2 3 4
– 1 0
– 4 – 8 – 6
1 –2
8 10 11
1 1 1 3 3 1
( )( ) = 2 4 7 5
( )
0 1 1 1 4 4 1 1 4 4
(ii) A shear with the x-axis as the invariant line and shear factor 1
(b) (i) The original rectangle
1
( – 1 2 4 7 5
)( ) = 1 3 3 1
0 1
( )
1 1 4 4 1 1 4 4
– 4 8 6
– 2 – 3 – 4
(ii) A shear with the x-axis as the invariant line and shear factor = – 1
20
IGCSE Revision Guide for Mathematics Model Answers © 2004, Hodder & Stoughton Educational