Question 1: The triangle A(1, 2), B(4, 4) and C(3, 7) is first reflected in the line y = 0 onto triangle A'B'C' and then \triangle A'B'C' is reflected in the origin onto \triangle A''B''C''. Write down the co-ordinates of :  

(i) A', B' \text{ and } C'           (ii) A'', B'' \text{ and } C''   

Answer:

\text{Reflection in y = 0 means reflection in x-axis. }

\text{(i) Since, reflection in the x-axis is given by } M_x (x, y) = ( x, -y) 

\text{Therefore } A' = \text{ reflection of } A (1, 2) \text{ in the x-axis } = ( 1, -2) 

\text{Similarly, } B' = (4, -4) \text{ and } C' = ( -3, 7) 

\text{(ii) Since reflection in the origin is given by } M_o (x, y) = ( -x, -y) 

\text{Therefore } A'' = \text{ reflection of } A' ( 1, -2) \text{ in the origin }= ( -1, 2) 

\text{Similarly, } B'' = ( -4, 4) \text{ and } C'' = ( -3, 7)

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Question 2: A point P is reflected in the x-axis. Co-ordinates of its image are (8, -6).

(i) Find the co-ordinates of P.

(ii) Find the co-ordinates of the image of P under reflection in the y-axis.

Answer:

\text{(i) Since, } M_x (8, 6) = (8, -6) \Rightarrow P = (8, 6)

\text{(ii) Co-ordinates of the image of P under reflection in the y-axis } = ( -8, 6) 

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\text{Question 3: Perform the operations } M_x.M_y \text{ and } M_y.M_x \text{ on the point } ( 3, -4). \\ \\ \text{ State whether it } M_x.M_y = M_y.M_x.  \text{ If 'yes'; then state whether it is always true. }

Answer:

M_x . M_y (3, -4) = M_x [ M_y \ ( 3, -4) ] = M_x \ (-3, -4) = ( -3, 4)

M_y . M_x ( 3, -4) =M_y [  M_x \ ( 3, -4) ] = M_y \ ( 3, 4) = ( 3, -4)

\therefore M_x.M_y = M_y.M_x

‘Yes’, it is always true.

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Question 4: Points (-5,0) and (4,0) are invariant points under reflection in the line L_1 ; points (0, – 6) and (0, 5) are invariant on reflection in the line L_2.

(a) Name or write equations for the lines L_1 and L_2.

(b) Write down the images of P(2, 6) and Q (-8, -3) on reflection in L_1. Name the images as P' and Q' respectively.

(c) Write down the images of P and Q on reflection in L_2. Name the images as P'' and Q''  respectively.

(d) State or describe a single transformation that maps Q'   onto Q''.

Answer:

(a) We know that every point in a line is invariant under the reflection in the same line.

Since, points (-5, 0) and (4,0) lie on the x-axis, therefore  Points (-5, 0) and (4, 0) are invariant under reflection in x-axis.

Given that the points (-5, 0) and (4,0) are invariant on reflection in line L_1.

Therefore The line L, is x-axis, whose equation is y = 0

Similarly, the given points (0, – 6) and (0, 5) lie on the y-axis and are invariant on reflection in line L_2.

Therefore The line L, is y-axis, whose equation is x = 0

(b) P’ = The image of P(2,6) in L_1  = The image of P(2,6) in x-axis = (2, -6)

Q’= The image of Q(-8, -3) in L_1  = The image of Q(-8, -3) in x-axis = (-8, 3)

(c) P” = The image of P(2,6) in L_2 = The image of P(2, 6) in y-axis = (-2, 6)

Q” = The image of Q(-8, -3) in L_2 = The image of Q(-8, -3) in y-axis = (8, -3)

(d) Since, Q’= (-8, 3) and Q” = (8, -3) and we know M_0(-x, y) = (x, -y)

Therefore the single transformation that maps Q’ onto Q” = Reflection in origin

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Question 5: (i) Find the reflection of the point P(-1, 3) in the line x = 2.
(ii) Find the reflection of the point Q(2, 1) in the line y + 3 = 0.

Answer:

(i) P(5, 3) is the reflection of P(-1, 3) in the line x=2

cx1

(ii) Q'(2, -7) is the reflection of Q(2, 1) in the line y+3 = 0

cx2

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Question 6: The points P(5, 1) and Q(-2, -2) are reflected in line x = 2. Use graph paper to find the images P’ and Q’ of points P and Q respectively in line x = 2. Take 2 cm equal to 2 units.

Answer:

The graph of line x =2 is the straight line AB, as shown below, which is parallel to y-axis and is at a distance of 2 units from it. Mark P(5, 1) and Q (-2, -2) on the same graph paper.

Mark P’ at the same distance behind AB as P is before it. Since P is 3 units before AB, its image. P’ will be 3 units behind AB. Clearly, the co-ordinates of P’ = (-1, 1).

In the same way, since Q(-2, -2) is 4 units before AB, its image Q’ will be 4 units behind AB. On marking position of Q’, we find : Q’ = (6, -2)

cx3

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Question 7: Use a graph paper for this question. (Take two divisions = I unit on both the axes). Plot the points P (3,2) and Q (-3, -2). From P and Q, draw perpendiculars PM and QN on the x-axis.
(a) Write the co-ordinates of points M and N.
(b) Name the image of P on reflection in the origin.
(c) Assign the special name to geometrical figure PMQN and find its area.
(d) Write the co-ordinates of the point to which M is mapped on reflection in:
(i) x-axis,        (ii) y-axis,         (iii) origin.  [ICSE Board 2003]

Answer:

(a) Co-ordinates of M = (3, 0) and Co-ordinates of N = (-3. 0)

(b) Image of P(3, 2) in origin = (-3, -2) = Q

(c) PMQN is a parallelogram

\displaystyle \text{Area of PMQN } = 2 ( \text{Area of } \triangle PMN) = 2 \frac{1}{2} \times 6 \times 2 = 12 \text{ sq. units. }

(d)

(i) M (3, 0) reflected in x-axis gives (3, 0)

(ii) M (3, 0) reflected in y-axis gives (-3, 0)

(iii) M (3, 0) reflected in origin gives (-3, 0)

cx4

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Question 8: Using the graph paper for this question.
The points A(2, 3), B(4, 5) and C(7, 2) arc the vertices of \triangle  ABC.
(i) Write down the coordinates of A’, B’, C’ if \triangle A' B' C'  is the image of \triangle  ABC, when reflected in the origin.
(ii) Write down the co-ordinates of A”, 8″, C” If \triangle  A" B" C"  is the image of \triangle  ABC, when reflected in the x-axis.
(iii) Mention the special name of the quadrilateral BCC”B” and find its area. [ ICSE Board 2006]

Answer:

\displaystyle \text{(i) A' = (-2, -3), B'(-4, -5) and C'= ( -7, -2) }

\displaystyle \text{(ii) A'' = ( 2, -3), B''=( 4, -5) and C''= ( 7, -2) }

\displaystyle \text{(iii) BCC''B'' is an isosceles trapezium as BB'' is parallel to CC'' and} \\ \\ \text{ BC = B''C'' }

\displaystyle \text{Area of quadrilateral } BCC''B'' = \frac{1}{2} ( BB'' + CC'') = \frac{1}{2} ( 10+4) \times 3 = 21 \text{ sq. units }

cx5