Questions


Given triangle ABC with vertices A (6, 5), B(4, 1) and C(3, 2) and that A(6, 5) is mapped onto A^{1}(6, 4) by a shear with yaxis in variant. On the grid provided below;
 draw triangle ABC
 draw triangle A^{1}B^{1}C^{1}, the image of triangle ABC, under the shear
 determine the matrix representing the shear

Triangle A^{1}B^{1}C^{1 }is mapped onto A^{11}B^{11}C^{11 }by a transformation defined by the matrix
 Draw triangle A^{11}B^{11}C^{11 }on the same grid as ABC and A^{1}B^{1}C^{1}
 Describe fully a single transformation that maps A^{11}B^{11}C^{11}

Given triangle ABC with vertices A (6, 5), B(4, 1) and C(3, 2) and that A(6, 5) is mapped onto A^{1}(6, 4) by a shear with yaxis in variant. On the grid provided below;


Under a certain rotation A( 2,0) is mapped onto A^{1}(4, 2) and B(0,5) is mapped onto B^{1}(9, 0)
 On the grid provided plot the lines AB and A^{1}B^{1 }on the same axes
 Hence determine by construction the coordinates of the centre and angle of rotation
 Under a quarter positive turn about the origin O, A^{1 }is mapped onto A^{11 }and B1 is mapped onto B^{11}. Determine the coordinates of A^{11 }and B^{11}
 Describe fully a single transformation which would map A to A^{11 }and B to B^{11}

Under a certain rotation A( 2,0) is mapped onto A^{1}(4, 2) and B(0,5) is mapped onto B^{1}(9, 0)

A transformation T is represented by the matrix and transformation U by the matrix. Given that a rectangle has coordinates at A (1,2) B(6, 2), C(6, 4) and D (1, 4) and that under T the image of ABCD is A^{1}B^{1}C^{1}D^{1} and under U the image of A_{1}B_{1}C_{1}D_{1} is A_{2}B_{2}C_{2}D_{2}:
 Find the coordinates of A_{1}B_{1}C_{1}D_{1 }and A_{2}B_{2}C_{2}D_{2}
 On the grid provided, plot ABCD, A_{1}B_{1}C_{1}D_{1 }and A_{2}B_{2}C_{2}D_{2}

Describe the transformation represented by:
 U
 UT
 If A_{2}B_{2}C_{2}D_{2 }were to be transformed by a transformation represented by the matrix to map onto A_{3}B_{3}C_{3}D_{3 }. What would be the area of A_{3}B_{3}C_{3}D_{3}

The vertices of a quadrilateral are A(2,2) B(8,2), C (8,6) and D(6,4) under a rotation the images of vertices A and D are A(0,8) and D^{1}(2, 12).
 On the grid provided and using the same axes draw the quadrilateral ABCD and the points A^{1 }and D^{1}
 Determine the centre and angle of rotation
 Locate the points B^{1 }and C^{1}under the rotation and complete the quadrilateral

A translation maps the point P(5, 3) onto P^{1}(2, 5)
 Determine the translation vector T
 A Point R^{1 }is the image of R(2, 3) under the same translation in (a) above, find the magnitude of P^{1}R^{1}

Triangle ABC has vertices at A(0, 1), B(4, 3)and C(2,2).
 Find the coordinates of image triangle A^{1}B^{1}C^{1} of triangle ABC under translation
 Given that triangle A^{11}B^{11}C^{11 }is the image of triangle A^{1}B^{1}C^{1 }under an enlargement scale factor 3, centre O(0,0) , find the coordinates of A^{11}, B^{11}and C^{11}
 If the area of triangle A^{1}B^{1}C^{1 }is 24 cm^{2}, calculate the area of triangle A^{11}B^{11}C^{11}
 Find the matrix that maps triangle A^{11}B^{11}C^{11 }onto triangle ABC

 The triangle ABC where A (2,1) B (1, 2) and C (4, 4) is reflected in the line X = 4 to give triangle A^{1}B^{1}C^{1}. Draw the two triangles on the graph provided and state the coordinates of A^{1}B^{1}C^{1}
 Draw the triangle A^{2 }(5,6), B^{2 }(2,7) and C^{2 }(0,4). Given that triangle A^{2}B^{2}C^{2 }is the image of triangle A^{1}B^{1}C^{1 }under rotation, determine the centre and angle of this rotation
 Show the image of triangle A^{2}B^{2}C^{2}, under an enlargement centre (0, 6) scale factor 1


Find the coordinates for the image of point P(6, 2) under the transformation defined by :
x^{1}= x – 3y
y^{1}= 2x 
 A quadrilateral ABCD has vertices A(4, 3), B(2, 3), C(4, 1) and D(5, 4). On the grid provided, draw the quadrilateral ABCD
 A^{1}B^{1}C^{1}D^{1 }is the image of ABCD under a rotation through +90^{o }about the origin. On the same axes, draw A^{1}B^{1}C^{1}D^{1 }under the transformation

A^{2}B^{2}C^{2}D^{2}is the image of under A^{1}B^{1}C^{1}D^{1 }under another transformation by the matrix
 Determine the coordinates of A^{2}B^{2}C^{2}D^{2}and plot it on the same axes
 Describe the transformation that maps A^{1}B^{1}C^{1}D^{1 }onto A^{2}B^{2}C^{2}D^{2}
 Find a single matrix of transformation that would map A^{2}B^{2}C^{2}D^{2 }onto ABCD

Find the coordinates for the image of point P(6, 2) under the transformation defined by :

 Triangle XYZ has vertices X(2, 1) Y(4, 1) and Z (4,2). Triangle XYZ maps onto triangle X^{1}Y^{1}Z^{1 }under transformation T1 = . Draw triangles XYZ and its image X^{1}Y^{1}Z^{1 }on the grid provided

Another triangle X^{11}Y^{11}Z^{11} is the image of X^{1}Y^{1}Z^{1} after transformation T_{2}
Draw triangle X^{11}Y^{11}Z^{11 }on the same set of axes  Find the single transformation matrix T that maps triangle XYZ on to the final image X^{11}Y^{11}Z^{11}
 Given that the area of triangle XYZ is 15cm^{2}, find the area of the triangle X^{11}Y^{11}Z^{11}

The quadrilateral A (2,1), B (4,1), C (4,4) and D (2,4) is mapped onto A’B’C’D’ by a matrix M_{1 }such that A^{1}(8,7), B^{1}(14,7), C^{1}(14,16) and D^{1}(8,16) .
 Draw both ABCD and A^{1}B^{1}C^{1}D^{1 }on the same plane
 Find the matrix of transformation that mapped ABCD onto A’B’C’D’ and describe it fully
 A^{1}B^{1}C^{1}D^{1 }underwent another matrix transformation at N which is a translation that gave the image A^{11}B^{11}C^{11}D^{11}, Where A^{11}(7,9), B^{11}(13,9), C^{11}(13,18) and D^{11}(7,18). The transformation N is a translation . Find the translation
 Draw A^{11}B^{11}C^{11}D^{11 }on the same axes where ABCD and A^{1}B^{1 }C^{1}D^{1 }were drawn

 On the grid provided. Plot the points A(2, 1) B (0, 3) C(2, 4) and D (4, 2) and join them to form a quadrilateral ABCD. What is the name of this quadrilateral?
 The points A^{1}(1, 2) B^{1}(3, 0) C^{1}(4, 2) and D^{1}(2, 4) are the images of ABC and D under a certain transformation T_{1}. On the same grid draw quadrilateral A^{1}B^{1}C^{1}D^{1 }and describe transformation T_{1 }fully.
 The points A^{11}(2, 4) B^{11}(6, 0) C1^{1}(8, 4) and D^{11}(4, 8) are the images of A^{1}B^{1}C^{1}D^{1}under transformation T2. On the same grid draw quadrilateral A^{11}B^{11}C^{11}D^{11 }and describe the transformation T_{2 }fully.
 On the same grid draw quadrilateral A^{111}B^{111}C^{111}D^{111}, the image of A^{11}B^{11}C^{11}D^{11 }under a reflection in the xaxis. State the coordinates of A^{111}B^{111}C^{111}D^{111}.

The Points A^{1}B^{1 }and C^{1 }are the images of A(4, 1), B( 0, 2) and C( 2, 4) respectively under a transformation represented by the matrix;
M= Write down the coordinates of A^{1}B^{1 }and C^{1}

A^{11}B^{11 }and C^{11 }are the images of A^{1}B^{1 }and C^{1}under another transformation whose matrix is:
N =
Write down the coordinates of A^{11}B^{11 }and C^{11} 
Transformation M followed by N can be represented by a single transformation P.
Determine the matrix for 
A matrix P is given by
Find P^{1}
 Triangle A^{1}B^{1}C^{1 }is the image of triangle ABC under a transformation represented by matrix T = If the area of triangle A^{1}B^{1}C^{1}is 25.6cm^{2}, find the area of the object

A point P(2, 4) is mapped into P^{1}(4, 0) under a translation.
Determine the image of point Q(1, 2) under the same translation 
The points A (2, 6), B (1, 1), C (2, 3) and D (4,0) are the vertices of quadrilateral ABCD.
 On graph paper plot the points A, B, C, and D and join them to form quadrilateral ABCD.
 The points A, B, C and D are the images of A^{1}, B^{1}, C^{1 }and D^{1 }respectively under an enlargement centre the origin and scale factor 2. On the same grid draw the image quadrilateral A^{1}B^{1}C^{1}D^{1}.
 The points A^{11}B^{11}C^{11 }and D11 are the images of ABCD respectively under reflection in the x – axis. On the same grid, locate the pints A^{11}B^{11}C^{11 }and D^{11 }and draw the second image quadrilateral A^{11}B^{11}C^{11}D^{11}.

Quadrilateral A^{111}B^{111}C^{111}D^{111 }is the image of ABCD under a certain transformation T.
Describe transformation T fully.
 T is a transformation represented by the matrix . Under T, a square of area 10cm^{2 }is mapped onto a square 110cm^{2}. Find the values of x
Answers


 B (4,5), C (3,6 ½ )
â ABC drawn
â ABC drawn  Shear maps
I (1, 1½ )
Matrix = 1 0
1 1½
 B (4,5), C (3,6 ½ )


 Half turn about (0,0)



 Centre (2, 2) 90^{o}
 A^{11} (2 , 4) , B^{11} (0, 9)
 Halfturn about the centre (0, 2)




 U   positive threequarter turn about the origin
 UT – Reflection I the line x = 0

IdetI = I2.5 x 2 – 1x 0 I= 5
∴ Area = 5x(5x2) = 20sq. units



 Centre (2, 4)
Angle +90°



A^{1}= (0+1, 12) = (1, 3)
B^{1}= (4 + 1) , 32) = (4, 1)
C^{1 }= ( 2 +1, 22) = (30)

Scale used S_{1}
ΔABC drawn B_{1}
ΔA_{1}B_{1}C_{1 }drawn B_{1}
A(6, 1), B(7, 2), C(4, 4) B_{1}
Line x = 4L_{1}
ΔA_{2 }B_{2 }C_{2 }drawn B_{1}
Two seen B_{1}
Centre of rotation
Angle of centre of rotation B_{1}
A_{3}B_{3}C_{3 }drawn B_{1}
Scale used S_{1}
ΔABC drawn B_{1}
ΔA_{1}B_{1}C_{1}drawn B_{1}
A, (6, 1), B(7, 2), C(4, 4) B_{1}
Line x = 4 L1
ΔA_{2}B_{2}C_{2}drawn B_{1}
Two seen B_{1}
Centre of rotation
Angle of centre of rotation B_{1}
ΔA_{3}B_{3}C_{3 }drawn B_{1} 

P(6, 2)
X^{1 }= 6 3 (2) = 12
Y^{1}= 2(6) = 12
(X^{1}, Y^{1}) = (12, 12) 
 A^{1}(3, 4)

B^{1 }(3, 2)
C^{1 }(1, 4)
D^{1}(4, 3)


P(6, 2)




ABCD drawn B_{1}
Name – Parallelogram B_{1}

A^{1}B^{1}C^{1}D^{1}drawn B_{1}
Attempt to joining any two points and bisecting. B^{1}
Description – Rotation + 900. B^{1 }or quarter turn about (0,0) 
A^{11}B^{11}C^{11}D^{11 }drawn. B_{1}
Description – Enlargement centre (0, 0) Scale factor –Z. B_{1} 
A^{111}B^{111}C^{111}D^{111 }– drawn. B_{1}
Attempt to reflect. B_{1}
Coordinates
A^{111 }= 92, 4) C^{111 }= (8, 4) B_{1}All correct
B^{111 }= (6, 0) D^{111 }(4, 8)

ABCD drawn B_{1}


Det = 2 – 6
=  4
A.S.F = 4
25.6 = 4
x
x = 6.4cm^{2}
Area of ΔABC = 6.4cm^{2} 
T + (2) = (4)
4 0
T = (4)  (2) = (2)
(0) (+4) (4)
∴ (2) + (1) = (1)
(4) (2) (6)
Q (1,6) 
5x^{2 }+ 6 = ^{110}/_{10}
5x^{2 }+ 6 = 11
x^{2}= 1
x = ±1