# Selina Concise Mathematics - Part II Solution for Class 10 Mathematics Chapter 12 - Reflection (In x-axis, y-axis, x=a, y=a and the origin ; Invariant Points)

## Selina Textbook Solutions Chapter 12 - Reflection (In x-axis, y-axis, x=a, y=a and the origin ; Invariant Points)

Selina Textbook Solutions are a perfect way to ace your examination with high marks. These Textbook Solutions are extremely helpful for solving difficult questions in the ICSE Class 10 Mathematics exam. Our Selina Textbook Solutions are written by our subject experts. Find all the answers to the Selina textbook questions of** **Chapter 12 - Reflection (In x-axis, y-axis, x=a, y=a and the origin ; Invariant Points).

All Selina textbook questions of** **Chapter 12 - Reflection (In x-axis, y-axis, x=a, y=a and the origin ; Invariant Points) solutions are created in accordance with the latest ICSE syllabus. These free Textbook Solutions for ICSE Class 10 Selina Concise Mathematics will give you a deeper insight on the fundamentals in this chapter and will help you to score more marks in the final examination. ICSE Class 10 students can refer to these solutions while doing their homework and while studying and revising for the Mathematics exam.

## Selina Concise Mathematics - Part II Solution for Class 10 Mathematics Chapter 12 - Reflection (In x-axis, y-axis, x=a, y=a and the origin ; Invariant Points) Page/Excercise 12(A)

Point |
Transformation |
Image |

(5, -7) |
Reflection in origin |
(-5, 7) |

(4, 2) |
Reflection in x-axis |
(4, -2) |

(0, 6) |
Reflection in y-axis |
(0, 6) |

(6, -6) |
Reflection in origin |
(-6, 6) |

(4, -8) |
Reflection in y-axis |
(-4, -8) |

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.

(i) (3, 2)

The co-ordinate of the given point under reflection in the x-axis is (3, -2).

(ii) (-5, 4)

The co-ordinate of the given point under reflection in the x-axis is (-5, -4).

(iii) (0, 0)

The co-ordinate of the given point under reflection in the x-axis is (0, 0).

(i) (6, -3)

The co-ordinate of the given point under reflection in the y-axis is (-6, -3).

(ii) (-1, 0)

The co-ordinate of the given point under reflection in the y-axis is (1, 0).

(iii) (-8, -2)

The co-ordinate of the given point under reflection in the y-axis is (8, -2).

(i) (-2, -4)

The co-ordinate of the given point under reflection in origin is (2, 4).

(ii) (-2, 7)

The co-ordinate of the given point under reflection in origin is (2, -7).

(iii) (0, 0)

The co-ordinate of the given point under reflection in origin is (0, 0).

(i) (-6, 4)

The co-ordinate of the given point under reflection in the line x = 0 is (6, 4).

(ii) (0, 5)

The co-ordinate of the given point under reflection in the line x = 0 is (0, 5).

(iii) (3, -4)

The co-ordinate of the given point under reflection in the line x = 0 is (-3, -4).

(i) (-3, 0)

The co-ordinate of the given point under reflection in the line y = 0 is (-3, 0).

(ii) (8, -5)

The co-ordinate of the given point under reflection in the line y = 0 is (8, 5).

(iii) (-1, -3)

The co-ordinate of the given point under reflection in the line y = 0 is (-1, 3).

(i) Since, M_{x} (-4, -5)
= (-4, 5)

So, the co-ordinates of P are (-4, -5).

(ii) Co-ordinates of the image of P under reflection in the y-axis (4, -5).

(i) Since, M_{O} (2, -7) =
(-2, 7)

So, the co-ordinates of P are (2, -7).

(ii) Co-ordinates of the image of P under reflection in the x-axis (2, 7).

M_{O} (a, b) = (-a, -b)

M_{y} (-a, -b) = (a, -b)

Thus, we get the co-ordinates of the point P' as (a, -b). It is given that the co-ordinates of P' are (4, 6).

On comparing the two points, we get,

a = 4 and b = -6

M_{x} (x, y) = (x, -y)

M_{O} (x, -y) = (-x, y)

Thus, we get the co-ordinates of the point P' as (-x, y). It is given that the co-ordinates of P' are (-8, 5).

On comparing the two points, we get,

x = 8 and y = 5

(i) The reflection in x-axis 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 y-axis is
given by M_{y} (x, y) = (-x, y).

The reflection of A (-3, 2) in y-axis is (3, 2).

Thus, the required single transformation is the reflection of A in the y-axis to the point 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 y-axis is given
by M_{y} (x, y) = (-x, y).

A'' = reflection of A' (-4, -6) in the y-axis = (4, -6)

(ii) The reflection in x-axis is
given by M_{x} (x, y) = (x, -y).

The reflection of A (4, 6) in x-axis is (4, -6).

Thus, the required single transformation is the reflection of A in the x-axis to the point A''.

(i) Reflection in y-axis is given
by M_{y} (x, y) = (-x, y)

A' = Reflection of A (2, 6) in y-axis = (-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 x-axis.

Reflection in x-axis is given by M_{x}
(x, y) = (x, -y)

P' = Reflection of P(-2, 3) in x-axis = (-2, -3)

Reflection in y-axis is given by M_{y}
(x, y) = (-x, y)

Q' = Reflection of Q(5, 4) in y-axis = (-5, 4)

Thus, the co-ordinates of points P' and Q' are (-2, -3) and (-5, 4) respectively.

The graph shows triangle ABC and triangle A'B'C' which is obtained when ABC is reflected in the origin.

(i) M_{x} . M_{y }(4,
-6) = M_{x} (-4, -6) = (-4, 6)

Single transformation equivalent
to M_{x} . M_{y} is M_{O}.

(ii) M_{y} . M_{x }(4,
-6) = M_{y} (4, 6) = (-4, 6)

Single transformation equivalent
to M_{y} . M_{x} is M_{O}.

(iii) M_{O} . M_{x }(4,
-6) = M_{O} (4, 6) = (-4, -6)

Single transformation equivalent
to M_{O} . M_{x} is M_{y}.

(iv) M_{x} . M_{O }(4,
-6) = M_{x} (-4, 6) = (-4, -6)

Single transformation equivalent
to M_{x} . M_{O} is M_{y}.

(v) M_{O} . M_{y }(4,
-6) = M_{O} (-4, -6) = (4, 6)

Single transformation equivalent
to M_{O} . M_{y} is M_{x}.

** **

(vi) M_{y} . M_{O }(4,
-6) = M_{y} (-4, 6) = (4, 6)

Single transformation equivalent
to M_{x} . M_{O} is M_{x}.

From (iii) and (iv), it is clear
that M_{O} . M_{x} = M_{x} . M_{O}.

From (v) and (vi), it is clear
that M_{y} . M_{O} = M_{O} . M_{y}.

Reflection in y-axis is given by M_{y}
(x, y) = (-x, y)

A' = Reflection of A(4, -1) in y-axis = (-4, -1)

Reflection in x-axis is given by M_{x}
(x, y) = (x, -y)

B' = Reflection of B in x-axis = (-2, 5)

Thus, B = (-2, -5)

(a) We know that reflection in the line x = 0 is the reflection in the y-axis.

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)

Co-ordinates of the image of (5, -8) in the line x = 0 are (-5, -8).

## Selina Concise Mathematics - Part II Solution for Class 10 Mathematics Chapter 12 - Reflection (In x-axis, y-axis, x=a, y=a and the origin ; Invariant Points) Page/Excercise 12(B)

(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 y-axis.

(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 x-axis.

So, (3, 0) and (-1, 0) are invariant under reflection in x-axis.

Hence,
the equation of line L_{1} is y = 0.

Similarly, (0, -3) and (0, 1) are invariant under reflection in y-axis.

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)
We know M_{x} (x, y) = (x, -y)

P' (5, -2) = reflection of P (a, b) in x-axis.

Thus, the co-ordinates of P are (5, 2).

Hence, a = 5 and b = 2.

(ii) P" = image of P (5, 2) reflected in y-axis = (-5, 2)

(iii) Single transformation that maps P' to P" is the reflection in origin.

(i) We know reflection of a point (x, y) in y-axis is (-x, y).

Hence, the point (-2, 0) when reflected in y-axis is mapped to (2, 0).

Thus, the mirror line is the y-axis and its equation is x = 0.

(ii) Co-ordinates of the image of (-8, -5) in the mirror line (i.e., y-axis) are (8, -5).

The line y = 3 is a line parallel to x-axis 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 co-ordinates 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 co-ordinates of Q' are (-2, 2).

The line x = 2 is a line parallel to y-axis 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 co-ordinates of P' are (6, 3).

A point P (a, b) is reflected in the x-axis to P' (2, -3).

We
know M_{x} (x, y) = (x, -y)

Thus, co-ordinates of P are (2, 3). Hence, a = 2 and b = 3.

P" = Image of P reflected in the y-axis = (-2, 3)

P''' = Reflection of P in the line (x = 4) = (6, 3)

(a) A' = Image of A under reflection in the x-axis = (3, -4)

(b) B' = Image of B under reflection in the line AA' = (6, 2)

(c) A" = Image of A under reflection in the y-axis = (-3, 4)

(d) B" = Image of B under reflection in the line AA" = (0, 6)

(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 x-axis = (3, -5)

(iii) C = Image of B when reflected in the y-axis = (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).

(a) Co-ordinates of P' = (-5, -3)

(b) Co-ordinates of M = (5, 0)

(c) Co-ordinates of N = (-5, 0)

(d) PMP'N is a parallelogram.

(e) Are of PMP'N = 2 (Area of D PMN)

(i) Co-ordinates 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'Q,

PO' =

So, perimeter of quadrilateral POP'O' = 4 PO' = 4 5 units = 20 units

(iv) Quadrilateral POP'O' is a rhombus.

Quadrilateral ABCD is an isosceles trapezium.

Co-ordinates 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.

(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 y-axis then it will remain invariant. Thus, the required axis is the y-axis.

(b) The co-ordinates of the image of Q (-2, 4) when reflected in y-axis 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) Co-ordinates of image of Q (-2, 4) when reflected in origin = (2, -4)

Co-ordinates of image of (2, -4) when reflected in x-axis = (2, 4)

Thus, the co-ordinates of the point are (2, 4).

(i) P (2, -4) is reflected in (x = 0) y-axis to get Q.

P(2, -4) Q (-2, -4)

(ii) Q (-2, -4) is reflected in (y = 0) x-axis to get R.

Q (-2, -4) R (-2, 4)

(iii) The figure PQR is right angled triangle.

(iv) Area of

(a)

i. A' = (4, 4) AND B' = (3, 0)

ii. The figure is an arrow head.

iii. The y-axis i.e. x = 0 is the line of symmetry of figure OABCB'A'.

## Selina Concise Mathematics X Class 10 Chapter Solutions

- Chapter 1 - Value Added Tax
- Chapter 2 - Banking (Recurring Deposit Accounts)
- Chapter 3 - Shares and Dividends
- Chapter 4 - Linear Inequations (in one variable)
- Chapter 5 - Quadratic Equations
- Chapter 6 - Solving (simple) Problmes (Based on Quadratic Equations)
- Chapter 7 - Ratio and Proportion (Including Properties and Uses)
- Chapter 8 - Remainder And Factor Theorems
- Chapter 9 - Matrices
- Chapter 10 - Arithmetic Progression
- Chapter 11 - Geometric Progression
- Chapter 12 - Reflection (In x-axis, y-axis, x=a, y=a and the origin ; Invariant Points)
- Chapter 13 - Section and Mid-Point Formula
- Chapter 14 - Equation of a Line
- Chapter 15 - Similarity (With Applications to Maps and Models)
- Chapter 16 - Loci (Locus and its Constructions)
- Chapter 17 - Circles
- Chapter 18 - Tangents and Intersecting Chords
- Chapter 19 - Constructions (Circles)
- Chapter 20 - Cylinder, Cone and Sphere (Surface Area and Volume)
- Chapter 21 - Trigonometrical Identities (Including Trigonometrical Ratios of Complementary Angles and Use of Four Figure Trigonometrical Tables)
- Chapter 22 - Heights and Distances
- Chapter 23 - Graphical Representation (Histograms and Ogives)
- Chapter 24 - Measures of Central Tendency (Mean, Median, Quartiles and Mode)
- Chapter 25 - Probability

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