SELINA Solutions for Class 10 Maths Chapter 12  Reflection (In xaxis, yaxis, x=a, y=a and the origin ; Invariant Points)
At TopperLearning, find complete Selina Solutions for ICSE Class 10 Mathematics Chapter 12 Reflection (In xaxis, yaxis, x = a, y = a and the origin; Invariant Points). Learn to note down the reflection of a point in the origin or the reflection of a point in a given line. Practise the solutions to understand how to answer questions on invariant points on reflection in a given line.
Further, revise the concept of single transformation by solving related problems from the textbook along with our Selena solutions. To get better at solving reflectionbased problems, check our ICSE Class 10 Maths videos, revision notes and other learning resources.
Chapter 12  Reflection (In xaxis, yaxis, x=a, y=a and the origin ; Invariant Points) Exercise Ex. 12(A)
Complete the following table:
Point 
Transformation 
Image 
(5, 7) 
(5, 7) 

(4, 2) 
Reflection in xaxis 

Reflection in yaxis 
(0, 6) 

(6, 6) 
(6, 6) 

(4, 8) 
(4, 8) 
Point 
Transformation 
Image 
(5, 7) 
Reflection in origin 
(5, 7) 
(4, 2) 
Reflection in xaxis 
(4, 2) 
(0, 6) 
Reflection in yaxis 
(0, 6) 
(6, 6) 
Reflection in origin 
(6, 6) 
(4, 8) 
Reflection in yaxis 
(4, 8) 
A point P is its own image under the reflection in a line l. Describe the position of point the P with respect to the line l.
Since, the point P is its own image under the reflection in the line l. So, point P is an invariant point.
Hence, the position of point P remains unaltered.
State the coordinates of the following points under reflection in xaxis:
(i) (3, 2)
(ii) (5, 4)
(iii) (0, 0)
(i) (3, 2)
The coordinate of the given point under reflection in the xaxis is (3, 2).
(ii) (5, 4)
The coordinate of the given point under reflection in the xaxis is (5, 4).
(iii) (0, 0)
The coordinate of the given point under reflection in the xaxis is (0, 0).
State the coordinates of the following points under reflection in yaxis:
(i) (6, 3)
(ii) (1, 0)
(iii) (8, 2)
(i) (6, 3)
The coordinate of the given point under reflection in the yaxis is (6, 3).
(ii) (1, 0)
The coordinate of the given point under reflection in the yaxis is (1, 0).
(iii) (8, 2)
The coordinate of the given point under reflection in the yaxis is (8, 2).
State the coordinates of the following points under reflection in origin:
(i) (2, 4)
(ii) (2, 7)
(iii) (0, 0)
(i) (2, 4)
The coordinate of the given point under reflection in origin is (2, 4).
(ii) (2, 7)
The coordinate of the given point under reflection in origin is (2, 7).
(iii) (0, 0)
The coordinate of the given point under reflection in origin is (0, 0).
State the coordinates of the following points under reflection in the line x = 0:
(i) (6, 4)
(ii) (0, 5)
(iii) (3, 4)
(i) (6, 4)
The coordinate of the given point under reflection in the line x = 0 is (6, 4).
(ii) (0, 5)
The coordinate of the given point under reflection in the line x = 0 is (0, 5).
(iii) (3, 4)
The coordinate of the given point under reflection in the line x = 0 is (3, 4).
State the coordinates of the following points under reflection in the line y = 0:
(i) (3, 0)
(ii) (8, 5)
(iii) (1, 3)
(i) (3, 0)
The coordinate of the given point under reflection in the line y = 0 is (3, 0).
(ii) (8, 5)
The coordinate of the given point under reflection in the line y = 0 is (8, 5).
(iii) (1, 3)
The coordinate of the given point under reflection in the line y = 0 is (1, 3).
A point P is reflected in the xaxis. Coordinates of its image are (4, 5).
(i) Find the coordinates of P.
(ii) Find the coordinates of the image of P under reflection in the yaxis.
(i) Since, M_{x} (4, 5) = (4, 5)
So, the coordinates of P are (4, 5).
(ii) Coordinates of the image of P under reflection in the yaxis (4, 5).
A point P is reflected in the origin. Coordinates of its image are (2, 7).
(i) Find the coordinates of P.
(ii) Find the coordinates of the image of P under reflection in the xaxis.
(i) Since, M_{O} (2, 7) = (2, 7)
So, the coordinates of P are (2, 7).
(ii) Coordinates of the image of P under reflection in the xaxis (2, 7).
The point (a, b) is first reflected in the origin and then reflected in the yaxis to P'. If P' has coordinates (4, 6); evaluate a and b.
M_{O} (a, b) = (a, b)
M_{y} (a, b) = (a, b)
Thus, we get the coordinates of the point P' as (a, b). It is given that the coordinates of P' are (4, 6).
On comparing the two points, we get,
a = 4 and b = 6
The point P (x, y) is first reflected in the xaxis and reflected in the origin to P'. If P' has coordinates (8, 5); evaluate x and y.
M_{x} (x, y) = (x, y)
M_{O} (x, y) = (x, y)
Thus, we get the coordinates of the point P' as (x, y). It is given that the coordinates of P' are (8, 5).
On comparing the two points, we get,
x = 8 and y = 5
The point A (3, 2) is reflected in the xaxis to the point A'. Point A' is then reflected in the origin to point A''.
(i) Write down the coordinates of A''.
(ii) Write down a single transformation that maps A onto A''.
(i) The reflection in xaxis is given by M_{x} (x, y) = (x, y).
A' = reflection of A (3, 2) in the x axis = (3, 2).
The reflection in origin is given by M_{O} (x, y) = (x, y).
A'' = reflection of A' (3, 2) in the origin = (3, 2)
(ii) The reflection in yaxis is given by M_{y} (x, y) = (x, y).
The reflection of A (3, 2) in yaxis is (3, 2).
Thus, the required single transformation is the reflection of A in the yaxis to the point A''.
The point A (4, 6) is first reflected in the origin to point A'. Point A' is then reflected in the yaxis to the point A''.
(i) Write down the coordinates of A''.
(ii) Write down a single transformation that maps A onto A''.
(i) The reflection in origin is given by M_{O} (x, y) = (x, y).
A' = reflection of A (4, 6) in the origin = (4, 6)
The reflection in yaxis is given by M_{y} (x, y) = (x, y).
A'' = reflection of A' (4, 6) in the yaxis = (4, 6)
(ii) The reflection in xaxis is given by M_{x} (x, y) = (x, y).
The reflection of A (4, 6) in xaxis is (4, 6).
Thus, the required single transformation is the reflection of A in the xaxis to the point A''.
The triangle ABC, where A is (2, 6), B is (3, 5) and C is (4, 7), is reflected in the yaxis to triangle A'B'C'. Triangle A'B'C' is then reflected in the origin to triangle A''B''C''.
(i) Write down the coordinates of A'', B'' and C''.
(ii) Write down a single transformation that maps triangle ABC onto triangle A''B''C''.
(i) Reflection in yaxis is given by M_{y} (x, y) = (x, y)
A' = Reflection of A (2, 6) in yaxis = (2, 6)
Similarly, B' = (3, 5) and C' = (4, 7)
Reflection in origin is given by M_{O} (x, y) = (x, y)
A'' = Reflection of A' (2, 6) in origin = (2, 6)
Similarly, B'' = (3, 5) and C'' = (4, 7)
(ii) A single transformation which maps triangle ABC to triangle A''B''C'' is reflection in xaxis.
P and Q have coordinates (2, 3) and (5, 4) respectively. Reflect P in the xaxis to P' and Q in the yaxis to Q'. State the coordinates of P' and Q'.
Reflection in xaxis is given by M_{x} (x, y) = (x, y)
P' = Reflection of P(2, 3) in xaxis = (2, 3)
Reflection in yaxis is given by M_{y} (x, y) = (x, y)
Q' = Reflection of Q(5, 4) in yaxis = (5, 4)
Thus, the coordinates of points P' and Q' are (2, 3) and (5, 4) respectively.
On a graph paper, plot the triangle ABC, whose vertices are at points A (3, 1), B (5, 0) and C (7, 4).
On the same diagram, draw the image of the triangle ABC under reflection in the origin O (0, 0).
The graph shows triangle ABC and triangle A'B'C' which is obtained when ABC is reflected in the origin.
Point A (4, 1) is reflected as A' in the yaxis. Point B on reflection in the xaxis is mapped as B' (2, 5). Write down the coordinates of A' and B.
Reflection in yaxis is given by M_{y} (x, y) = (x, y)
A' = Reflection of A(4, 1) in yaxis = (4, 1)
Reflection in xaxis is given by M_{x} (x, y) = (x, y)
B' = Reflection of B in xaxis = (2, 5)
Thus, B = (2, 5)
The point (5, 0) on reflection in a line is mapped as (5, 0) and the point (2, 6) on reflection in the same line is mapped as (2, 6).
(a) Name the line of reflection.
(b) Write down the coordinates of the image of (5, 8) in the line obtained in (a).
(a) We know that reflection in the line x = 0 is the reflection in the yaxis.
It is given that:
Point (5, 0) on reflection in a line is mapped as (5, 0).
Point (2, 6) on reflection in the same line is mapped as (2, 6).
Hence, the line of reflection is x = 0.
(b) It is known that M_{y} (x, y) = (x, y)
Coordinates of the image of (5, 8) in the line x = 0 are (5, 8).
Chapter 12  Reflection (In xaxis, yaxis, x=a, y=a and the origin ; Invariant Points) Exercise Ex. 12(B)
Attempt this question on graph paper.
(a) Plot A (3, 2) and B (5, 4) on graph paper. Take 2 cm = 1 unit on both the axes.
(b) Reflect A and B in the xaxis to A' and B' respectively. Plot these points also on the same graph paper.
(c) Write down:
(i) the geometrical name of the figure ABB'A';
(ii) the measure of angle ABB';
(iii) the image of A'' of A, when A is reflected in the origin.
(iv) the single transformation that maps A' to A''.
(c)
(i) From graph, it is clear that ABB'A' is an isosceles trapezium.
(ii) The measure of angle ABB' is 45°.
(iii) A'' = (3, 2)
(iv) Single transformation that maps A' to A" is the reflection in yaxis.
Points (3, 0) and (1, 0) are invariant points under reflection in the line L_{1}; points (0, 3) and (0, 1) are invariant points on reflection in line L_{2}.
(i) Name or write equations for the lines L_{1} and L_{2}.
(ii) Write down the images of the points P (3, 4) and Q (5, 2) on reflection in line L_{1}. Name the images as P' and Q' respectively.
(iii) Write down the images of P and Q on reflection in L_{2}. Name the images as P" and Q" respectively.
(iv) State or describe a single transformation that maps P' onto P".
(i) We know that every point in a line is invariant under the reflection in the same line.
Since points (3, 0) and (1, 0) lie on the xaxis.
So, (3, 0) and (1, 0) are invariant under reflection in xaxis.
Hence, the equation of line L_{1} is y = 0.
Similarly, (0, 3) and (0, 1) are invariant under reflection in yaxis.
Hence, the equation of line L_{2} is x = 0.
(ii) P' = Image of P (3, 4) in L_{1} = (3, 4)
Q' = Image of Q (5, 2) in L_{1} = (5, 2)
(iii) P'' = Image of P (3, 4) in L_{2} = (3, 4)
Q'' = Image of Q (5, 2) in L_{2} = (5, 2)
(iv) Single transformation that maps P' onto P" is reflection in origin.
(i) Point P (a, b) is reflected in the xaxis to P' (5, 2). Write down the values of a and b.
(ii) P" is the image of P when reflected in the yaxis. Write down the coordinates of P".
(iii) Name a single transformation that maps P' to P".
(i) We know M_{x} (x, y) = (x, y)
P' (5, 2) = reflection of P (a, b) in xaxis.
Thus, the coordinates of P are (5, 2).
Hence, a = 5 and b = 2.
(ii) P" = image of P (5, 2) reflected in yaxis = (5, 2)
(iii) Single transformation that maps P' to P" is the reflection in origin.
The point (2, 0) on reflection in a line is mapped to (2, 0) and the point (5, 6) on reflection in the same line is mapped to (5, 6).
(i) State the name of the mirror line and write its equation.
(ii) State the coordinates of the image of (8, 5) in the mirror line.
(i) We know reflection of a point (x, y) in yaxis is (x, y).
Hence, the point (2, 0) when reflected in yaxis is mapped to (2, 0).
Thus, the mirror line is the yaxis and its equation is x = 0.
(ii) Coordinates of the image of (8, 5) in the mirror line (i.e., yaxis) are (8, 5).
The points P (4, 1) and Q (2, 4) are reflected in line y = 3. Find the coordinates of P', the image of P and Q', the image of Q.
The line y = 3 is a line parallel to xaxis and at a distance of 3 units from it.
Mark points P (4, 1) and Q (2, 4).
From P, draw a straight line perpendicular to line CD and produce. On this line mark a point P' which is at the same distance above CD as P is below it.
The coordinates of P' are (4, 5).
Similarly, from Q, draw a line perpendicular to CD and mark point Q' which is at the same distance below CD as Q is above it.
The coordinates of Q' are (2, 2).
A point P (2, 3) is reflected in line x = 2 to point P'. Find the coordinates of P'.
The line x = 2 is a line parallel to yaxis and at a distance of 2 units from it.
Mark point P (2, 3).
From P, draw a straight line perpendicular to line CD and produce. On this line mark a point P' which is at the same distance to the right of CD as P is to the left of it.
The coordinates of P' are (6, 3).
A point P (a, b) is reflected in the xaxis to P' (2, 3). Write down the values of a and b. P" is the image of P, reflected in the yaxis. Write down the coordinates of P". Find the coordinates of P''', when P is reflected in the line, parallel to yaxis, such that x = 4.
A point P (a, b) is reflected in the xaxis to P' (2, 3).
We know M_{x} (x, y) = (x, y)
Thus, coordinates of P are (2, 3). Hence, a = 2 and b = 3.
P" = Image of P reflected in the yaxis = (2, 3)
P''' = Reflection of P in the line (x = 4) = (6, 3)
Points A and B have coordinates (3, 4) and (0, 2) respectively. Find the image:
(a) A' of A under reflection in the xaxis.
(b) B' of B under reflection in the line AA'.
(c) A" of A under reflection in the yaxis.
(d) B" of B under reflection in the line AA".
(a) A' = Image of A under reflection in the xaxis = (3, 4)
(b) B' = Image of B under reflection in the line AA' = (6, 2)
(c) A" = Image of A under reflection in the yaxis = (3, 4)
(d) B" = Image of B under reflection in the line AA" = (0, 6)
(i) Plot the points A (3, 5) and B (2, 4). Use 1 cm = 1 unit on both the axes.
(ii) A' is the image of A when reflected in the xaxis. Write down the coordinates of A' and plot it on the graph paper.
(iii) B' is the image of B when reflected in the yaxis, followed by reflection in the origin. Write down the coordinates of B' and plot it on the graph paper.
(iv) Write down the geometrical name of the figure AA'BB'.
(v) Name the invariant points under reflection in the xaxis.
(i) The points A (3, 5) and B (2, 4) can be plotted on a graph as shown.
(ii) A' = Image of A when reflected in the xaxis = (3, 5)
(iii) C = Image of B when reflected in the yaxis = (2, 4)
B' = Image when C is reflected in the origin = (2, 4)
(iv) Isosceles trapezium
(v) Any point that remains unaltered under a given transformation is called an invariant.
Thus, the required two points are (3, 0) and (2, 0).
The point P (5, 3) was reflected in the origin to get the image P'.
(a) Write down the coordinates of P'.
(b) If M is the foot if the perpendicular from P to the xaxis, find the coordinates of M.
(c) If N is the foot if the perpendicular from P' to the xaxis, find the coordinates of N.
(d) Name the figure PMP'N.
(e) Find the area of the figure PMP'N.
(a) Coordinates of P' = (5, 3)
(b) Coordinates of M = (5, 0)
(c) Coordinates of N = (5, 0)
(d) PMP'N is a parallelogram.
(e) Are of PMP'N = 2 (Area of D PMN)
The point P (3, 4) is reflected to P' in the xaxis; and O' is the image of O (the origin) when reflected in the line PP'. Write:
(i) the coordinates of P' and O'.
(ii) the length of the segments PP' and OO'.
(iii) the perimeter of the quadrilateral POP'O'.
(iv) the geometrical name of the figure POP'O'.
(i) Coordinates of P' and O' are (3, 4) and (6, 0) respectively.
(ii) PP' = 8 units and OO' = 6 units.
(iii) From the graph it is clear that all sides of the quadrilateral POP'O' are equal.
In right PO'O,
PO' =
So, perimeter of quadrilateral POP'O' = 4 PO' = 4 5 units = 20 units
(iv) Quadrilateral POP'O' is a rhombus.
A (1, 1), B (5, 1), C (4, 2) and D (2, 2) are vertices of a quadrilateral. Name the quadrilateral ABCD. A, B, C, and D are reflected in the origin on to A', B', C' and D' respectively. Locate A', B', C' and D' on the graph sheet and write their coordinates. Are D, A, A' and D' collinear?
Quadrilateral ABCD is an isosceles trapezium.
Coordinates of A', B', C' and D' are A'(1, 1), B'(5, 1), C'(4, 2) and D'(2, 2) respectively.
It is clear from the graph that D, A, A' and D' are collinear.
P and Q have coordinates (0, 5) and (2, 4).
(a) P is invariant when reflected in an axis. Name the axis.
(b) Find the image of Q on reflection in the axis found in (a).
(c) (0, k) on reflection in the origin is invariant. Write the value of k.
(d) Write the coordinates of the image of Q, obtained by reflecting it in the origin followed by reflection in xaxis.
(a) Any point that remains unaltered under a given transformation is called an invariant.
It is given that P (0, 5) is invariant when reflected in an axis. Clearly, when P is reflected in the yaxis then it will remain invariant. Thus, the required axis is the yaxis.
(b) The coordinates of the image of Q (2, 4) when reflected in yaxis is (2, 4).
(c) (0, k) on reflection in the origin is invariant. We know the reflection of origin in origin is invariant. Thus, k = 0.
(d) Coordinates of image of Q (2, 4) when reflected in origin = (2, 4)
Coordinates of image of (2, 4) when reflected in xaxis = (2, 4)
Thus, the coordinates of the point are (2, 4).
(i) The point P (2, 4) is reflected about the line x = 0 to get the image Q. Find the coordinates of Q.
(ii) The point Q is reflected about the line y = 0 to get the image R. Find the coordinates or R.
(iii) Name the figure PQR.
(iv) Find the area of figure PQR.
(i) P (2, 4) is reflected in (x = 0) yaxis to get Q.
P(2, 4) Q (2, 4)
(ii) Q (2, 4) is reflected in (y = 0) xaxis to get R.
Q (2, 4) R (2, 4)
(iii) The figure PQR is right angled triangle.
(iv) Area of
Using a graph paper, plot the point A (6, 4) and B (0, 4).
(a) Reflect A and B in the origin to get the image A' and B'.
(b) Write the coordinates of A' and B'.
(c) Sate the geometrical name for the figure ABA'B'.
(d) Find its perimeter.
(a)
Use graph paper for this question.
(Take 2 cm = 1 unit along both side xaxis and yaxis.)
Plot the points O(0,0), A(4, 4), B(3, 0) and C(0, 3).
i. Reflect points A and B on the yaxis and name them A' and B' respectively. Write down their coordinates.
ii. Name the figure OABCB'A'.
iii. State the line of symmetry of this figure. [Now symmetry is not in course]
i. A' = (4, 4) AND B' = (3, 0)
ii. The figure is an arrow head.
iii. The yaxis i.e. x = 0 is the line of symmetry of figure OABCB'A'.
Use a graph paper for this question: (Take 2cm = 1 unit on both x and y axes) (i) Plot the following points: A(0,4), B(2,3), C(1,1) and D(2,0).
(ii) Reflect points B, C, D on the yaxis and write down their coordinates. Name the images as B', C', D' respectively.
(iii) Join the points A, B, C, D, D', C', B' and A in order, so as to form a closed figure. Write down the equation to the line about which if this closed figure obtained is folded, the two parts of the figure exactly coincide.
(i)Plotting A(0,4), B(2,3), C(1,1) and D(2,0).
(ii) Reflected points B'(2,3), C'(1,1) and D'(2,0).
(iii) The figure is symmetrical about x = 0
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