J20

(c) Describe fully the single transformation that maps PQRS onto P2Q2R2S2.
(d) Find the matrix of transformation that maps PQRS onto P2Q2R2S2.
3. Draw triangle T whose vertices are (4, 3), (-3, -1) and (-1, 3).
(a) Triangle R is the image of triangle T under the transformation represented
0.6 0.8
by the matrix N =
0.8 0.6
(i) Calculate the coordinates of the vertices of triangle R.
(ii) Draw and label triangle R
(b) Describe fully a single transformation that maps triangle T onto triangle R.
4. Triangle XYZ is such that X(1, 4), Y(1, 7) and Z(3, 1) are its vertices.
Triangle XYZ is mapped onto triangle X 1Y 1Z 1 by the transformation
0
1
represented by the matrix M = .
1
0
(a) Draw and label triangle X 1Y 1Z 1 and describe fully the transformation that maps
triangle XYZ onto X 1Y 1Z 1 in geometrical terms.
(b) Reflect ∆XYZ in the y-axis, and label the vertices of its image as X2, Y2, and Z2.
Write down the elements of matrix N which represents this transformation.
(c) Triangle X1Y1Z1 can be mapped onto triangle X2Y2Z2 by a single transformation
P.
(i) Describe this transformation fully.
(ii) Write down the elements of matrix P which represents this
transformation.
(iii) State the relationship between M, N and P.
The inverse of a transformation
The inverse of a transformation reverses the transformation, i.e. it is the transformation
which takes the image back to the object. For example, the inverse of an anticlockwise
rotation of 90 0 about O is a clockwise rotation of 90 0 about O.
3
If translation T has vector , the translation which has the opposite effect has vector
2
3
. This is written as T
-1
2
If M is a transformation representing a transformation, then the matrix representing the
inverse of the transformation is indicated as M -1 .
Example
1
2
Matrix N = describes a transformation on ∆PQR. The coordinates of the image
2 3
are P1(-3, 2), Q1(2, 1) and R1(0, 1). Find:
(a) the matrix that maps ∆P1Q1R1 onto ∆PQR.
(b) the coordinates of P, Q and R.