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) $A'', B'' \text{ and } C''$

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

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

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

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

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

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

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

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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.

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

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

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

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

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

$\therefore M_x.M_y = M_y.M_x$

‘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 $Q''.$

(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.

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

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

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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.

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)

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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]

(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)

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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]

$\displaystyle \text{(i) A' = (-2, -3), B'(-4, -5) and C'= ( -7, -2) }$
$\displaystyle \text{(ii) A'' = ( 2, -3), B''=( 4, -5) and C''= ( 7, -2) }$
$\displaystyle \text{(iii) BCC''B'' is an isosceles trapezium as BB'' is parallel to CC'' and} \\ \\ \text{ BC = B''C'' }$
$\displaystyle \text{Area of quadrilateral } BCC''B'' = \frac{1}{2} ( BB'' + CC'') = \frac{1}{2} ( 10+4) \times 3 = 21 \text{ sq. units }$