Class 10 SELINA Solutions Maths Chapter 12 - Reflection (In x-axis, y-axis, x=a, y=a and the origin ; Invariant Points)

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-ordinates of the given point under reflection in the x-axis are (3, -2).

(ii) (-5, 4)

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

(iii) (0, 0)

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

(i) (6, -3)

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

(ii) (-1, 0)

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

(iii) (-8, -2)

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

(i) (-2, -4)

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

(ii) (-2, 7)

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

(iii) (0, 0)

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

(i) (-6, 4)

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

(ii) (0, 5)

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

(iii) (3, -4)

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

(i) (-3, 0)

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

(ii) (8, -5)

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

(iii) (-1, -3)

The co-ordinates of the given point under reflection in the line y = 0 are (-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 are (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 triangle ABC is reflected in the origin.

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, that is y-axis.

(b) It is known that M_y (x, y) = (-x, y)

Co-ordinates of the image of (5, -8) in y-axis are (-5, -8).

(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 L₁ is y = 0.

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

Hence, the equation of line L₂ is x = 0.

(ii) P' = Image of P (3, 4) in L₁ = (3, -4)

Q' = Image of Q (-5, -2) in L₁ = (-5, 2)

(iii) P'' = Image of P (3, 4) in L₂ = (-3, 4)

Q'' = Image of Q (-5, -2) in L₂ = (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).

And, the point (5, -6) when reflected in y-axis is mapped to (-5, -6).

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 CD parallel to x-axis and at a distance of 3 units from it.

Mark points P(4, 1) and Q(-2, 4).

Mark P' at the same distance above CD as P is below it. Since, P is 2 units below CD, its image P' will be 2 units above CD.

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

Similarly, mark Q' at the same distance below CD as Q is above it. Since, Q is 1 unit above CD, its image Q' will be 1 unit below CD.

Hence, the co-ordinates of Q' are (-2, 2).

The line x = 2 is a line CD parallel to y-axis and at a distance of 2 units from it.

Mark point P(-2, 3).

Mark P' at the same distance to the right of CD as P is to the left of it.

Hence, 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) Area 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 PQO',

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).

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

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

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

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

(c) The figure PQR is right angled triangle.

(d) 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'.

(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