Class 10: Coordinate Geometry (Reflections)

$\displaystyle \textbf{Question 1: } \text{The triangle }A(1,2),\ B(4,4)\text{ and }C(3,7)\text{ is first reflected in the line }$
$\displaystyle y=0 \ \text{onto }\triangle A'B'C'\text{ and then }\triangle A'B'C'\text{ is reflected in the origin onto }\triangle A''B''C''.$
$\displaystyle \text{Write down the coordinates of:}$
$\displaystyle \text{(i) }A',\ B'\text{ and }C' \qquad \text{(ii) }A'',\ B''\text{ and }C''$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Reflection in }y=0\text{ means reflection in the x-axis.}$
$\displaystyle \text{(i) Reflection in the x-axis is given by }M_x(x,y)=(x,-y).$
$\displaystyle \therefore A'=(1,-2),\ B'=(4,-4)\text{ and }C'=(3,-7).$
$\displaystyle \text{(ii) Reflection in the origin is given by }M_O(x,y)=(-x,-y).$
$\displaystyle \therefore A''=(-1,2),\ B''=(-4,4)\text{ and }C''=(-3,7).$
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$\displaystyle \textbf{Question 2: } \text{A point }P\text{ is reflected in the x-axis. Coordinates of its image are }(8,-6).$
$\displaystyle \text{(i) Find the coordinates of }P. \qquad \\ \text{(ii) Find the coordinates of the image of }P\text{ under reflection in the y-axis.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) Reflection in the x-axis is given by }M_x(x,y)=(x,-y).$
$\displaystyle \text{Since }M_x(8,6)=(8,-6),\text{ the coordinates of }P=(8,6).$
$\displaystyle \text{(ii) Reflection of }P(8,6)\text{ in the y-axis is }(-8,6).$
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$\displaystyle \textbf{Question 3: } \text{Perform the operations }M_x.M_y\text{ and }M_y.M_x\text{ on the point }(3,-4).$
$\displaystyle \text{State whether }M_x.M_y=M_y.M_x.\text{ If yes, state whether it is always true.}$
$\displaystyle \text{Answer:}$
$\displaystyle M_x.M_y(3,-4)=M_x\!\left[M_y(3,-4)\right]=M_x(-3,-4)=(-3,4)$
$\displaystyle M_y.M_x(3,-4)=M_y\!\left[M_x(3,-4)\right]=M_y(3,4)=(-3,4)$
$\displaystyle \therefore M_x.M_y=M_y.M_x=(-3,4).$
$\displaystyle \text{Yes, }M_x.M_y=M_y.M_x\text{ and this is always true.}$
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$\displaystyle \textbf{Question 4: } \text{Points }(-5,0)\text{ and }(4,0)\text{ are invariant points under reflection in the line }L_1.$
$\displaystyle \text{Points }(0,-6)\text{ and }(0,5)\text{ are invariant under reflection in the line }L_2.$
$\displaystyle \text{(a) Name or write equations for the lines }L_1\text{ and }L_2.$
$\displaystyle \text{(b) Write the images of }P(2,6)\text{ and }Q(-8,-3)\text{ on reflection in }L_1.$
$\displaystyle \text{(c) Write the images of }P\text{ and }Q\text{ on reflection in }L_2.$
$\displaystyle \text{(d) State a single transformation that maps }Q'\text{ onto }Q''.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{(a) Points }(-5,0)\text{ and }(4,0)\text{ lie on the x-axis.}$
$\displaystyle \therefore L_1\text{ is the x-axis, whose equation is }y=0.$
$\displaystyle \text{Points }(0,-6)\text{ and }(0,5)\text{ lie on the y-axis.}$
$\displaystyle \therefore L_2\text{ is the y-axis, whose equation is }x=0.$
$\displaystyle \text{(b) Reflection in }L_1\text{ means reflection in the x-axis.}$
$\displaystyle \therefore P'=(2,-6)\text{ and }Q'=(-8,3).$
$\displaystyle \text{(c) Reflection in }L_2\text{ means reflection in the y-axis.}$
$\displaystyle \therefore P''=(-2,6)\text{ and }Q''=(8,-3).$
$\displaystyle \text{(d) Since }Q'=(-8,3)\text{ and }Q''=(8,-3),$
$\displaystyle \text{the single transformation that maps }Q'\text{ onto }Q''\text{ is reflection in the origin.}$
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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

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

(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]