Reflection

Exercise 2

Question 1

Find the co-ordinates of the images of the following points under reflection in the x- axis:
(i) (2, -5)
(ii) (-3/2, -1/2)
(iii) (-7, 0)

Solution

Given:

Reflection is in the x - axis.
∴ For any point P(x,y), its reflection P' will be P'(x, -y)

(i) (2, -5) will be (2,5)
(ii) (-3/2, -1/2) will be (-3/2,1/2)
(iii) (-7, 0) will be (-7,0)

Question 2

Find the co-ordinates of the images of the following points under reflection in the y- axis:
(i) (2, -5)
(ii) (-3/2, 1/2)
(iii) (0, -7)

Solution

Given:

Reflection is in the y - axis.
∴ For any point P(x,y), its reflection P' will be P'(-x, y)

(i) (2, -5) will be (-2,-5)
(ii) (-3/2, 1/2) will be (3/2,1/2)
(iii) (0, -7) will be (0,-7)

Question 3

Find the co-ordinates of the images of the following points under reflection in the origin:
(i) (2, -5)
(ii) (-3/2, -1/2)
(iii) (0, 0)

Solution

Given:

Reflection is in the origin.
∴ For any point P(x,y), its reflection P' will be P'(-x, -y)

(i) (2, -5) will be (-2,5)
(ii) (-3/2, -1/2) will be (3/2,1/2)
(iii) (0, 0) will be (0,0)

Question 4

The image of a point P under reflection in the x-axis is (5, -2). Write down the coordinates of P.

Solution

Given:

Reflection is in the x - axis.
∴ For any point P(x,y), its reflection P' will be P'(x, -y)
Here P'(x,-y) = (5, -2)

∴ Coordinates of P = P(x,y) = (5,2)

Question 5

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.

Solution

(i) Reflection is in the x - axis.
∴ For any point P(x,y), its reflection P' will be P'(x, -y) Here P'(x,-y) = (8, -6)

∴ Coordinates of P = P(x,y) = (8,6)

(i) Reflection is in the y - axis.
∴ For any point P(x,y), its reflection P' will be P'(-x, y) Here P(x,y) = (8,6)

∴ Coordinates of its image P' = P'(-x,y) = (-8,6)

Question 6

A point P is reflected in the origin. Co-ordinates of its image are (2, -5). Find
(i) the co-ordinates of P.
(ii) the co-ordinates of the image of P in the x-axis.

Solution

(i) Reflection is in the origin.
∴ For any point P(x,y), its reflection P' will be P'(-x, -y) Here P'(-x,-y) = (2, -5)

∴ Coordinates of P = P(x,y) = (-2,5)

(ii) Reflection is in the x - axis.
∴ For any point P(x,y), its reflection P' will be P'(x, -y) Here P(x,y) = (-2,5)

∴ Coordinates of its image P' = P'(x,-y) = (-2,-5)

Question 7

(i) The point P (2, 3) is reflected in the line x = 4 to the point P’. Find the co-ordinates of the point P’.

(ii) Find the image of the point P (1, -2) in the line x = -1.

Solution

(i) Graph:

Explanation:

Take 1cm = 1 unit on both axes.
Plot the line x = 4 that will cross X axis at point (4,0) and will be parallel to Y axis.
Plot the point P(2,3) on the graph.
∴ Its image on reflection in the line x=4 will be P' = (6,3)

(ii) Image of the point P (1, -2) in the line x = -1.

Graph:

Explanation:

Take 1cm = 1 unit on both axes.
Plot the line x = -1 that will cross x axis at point (-1,0) and will be parallel to Y axis.
Plot the point P(1,-2) on the graph.
∴ Its image on reflection in the line x = -1 will be P' = (-3,-2)

Question 8

(i) The point P(2, 4) on reflection in the line y = 1 is mapped onto P’ Find the co-ordinates of P’.

(ii) Find the image of the point P(-3, -5) in the line y = -2.

Solution

(i) Graph:

Explanation:

Take 1cm = 1 unit on both axes.
Plot the line y = 1 that will cross x axis at point (0,1) and will be parallel to y axis.
Plot the point P(2,4) on the graph.
∴ Its image on reflection in the line y = 1 will be P' = (2,-2)

(ii) The image of the point P(-3, -5) in the line y = -2.

Graph:

Explanation:

Take 1cm = 1 unit on both axes.
Plot the line y = -2 that will cross x axis at point (0,-2) and will be parallel to y axis.
Plot the point P(-3,-5) on the graph.
∴ Its image on reflection in the line x = -1 will be P' = (-3,1)

Question 9

The point P (-4, -5) on reflection in y-axis is mapped on P’. The point P’ on reflection in the origin is mapped on P”. Find the co-ordinates of P’ and P”. Write down a single transformation that maps P onto P”.

Solution

Given:

First reflection is in the y - axis.
For any point P(x,y), its reflection P' in the y - axis will be P'(-x,y)
Since P(x,y) = P(-4,-5)
∴ Coordinates of P' = P(-x,y) = (4, -5)

Second reflection is in the origin. For any point P(x,y), its reflection in the origin will be P'(-x,-y)
Since P(x,y) in this case = P(4,-5)
∴ Coordinates of P'' = P(-x,-y) = (-4, 5)

On comparing P(-4,-5) to P''(-4,5) we see that sign of only Y coordinate is changing.

∴ the single transformation that maps P to P'' is Reflection in the x - axis

Question 10

Write down the co-ordinates of the image of the point (3, -2) when:
(i) reflected in the x-axis
(ii) reflected in the y-axis
(iii) reflected in the x-axis followed by a reflection in the y-axis
(iv) reflected in the origin.

Solution

(i) Reflection is in the x - axis.
For any point P(x,y), its reflection P' in the x-axis will be P'(x,-y)
Since P(x,y) = P(3,-2)
∴ Coordinates of P' = P(x,-y) = (3,2)

(ii) Reflection is in the y - axis.
For any point P(x,y), its reflection P' in the y-axis will be P'(-x,y)
Since P(x,y) = P(3,-2)
∴ Coordinates of P' = P(-x,y) = (-3,-2)

(iii) First reflection is in the x - axis.
For any point P(x,y), its reflection P' in the x-axis will be P'(x,-y)
Since P(x,y) = P(3,-2)
∴ Coordinates of P' = P(x,-y) = (3,2)

Second reflection is in the y-axis. For any point P(x,y), its reflection in the y-axis will be P'(-x,y)
Since P(x,y) in this case = P(3,2)
∴ Coordinates of P'' = P(-x,y) = (-3, 2)

(iv) Reflection is in the origin.
For any point P(x,y), its reflection P' in the origin will be P'(-x,-y)
Since P(x,y) = P(3,-2)
∴ Coordinates of P' = P(-x,-y) = (-3,2)

Question 11

Find the co-ordinates of the image of (3, 1) under reflection in x-axis followed by a reflection in the line x = 1.

Solution

Graph:

Explanation:

First reflection is in the x - axis.
For any point P(x,y), its reflection P' in the x-axis will be P'(x,-y)
Since P(x,y) = P(3,1)
∴ Coordinates of P' = P'(x,-y) = (3,-1)

Take 1cm = 1 unit on both axes.
Plot the line x = 1 that will cross x axis at point (1,0) and will be parallel to y axis.
Plot the point P(3,1) and P'(3,-1) on the graph.
∴ Its image on reflection in the line x = 1 will be P'' = (-1,-1)

Question 12

If P’(-4, -3) is the image of a point P under reflection in the origin, find (i) the co-ordinates of P. (ii) the co-ordinates of the image of P under reflection in the line y = -2.

Solution

(i) For any point P(x,y), its reflection P' in the origin will be P'(-x,-y)
Since P'(-x,-y) = P'(-4,-3)
∴ Coordinates of P = P(x,y) = (4,3)

(ii) Graph:

Explanation:

Take 1cm = 1 unit on both axes.
Plot the line y = -2 that will cross y-axis at point (0,-2) and will be parallel to x-axis.
Plot the point P(4,3) and P'(-4,-3) on the graph.
From Graph: Reflection of P(4,3) on line y= -2 is P'' = (4,-7)

Question 13

A point P (a, b) is reflected in the x-axis to P’ (2, -3), write down the values of a and b. P” is the image of P, when reflected in the y-axis. Write down the co-ordinates of P”. Find the co-ordinates of P”, when P is reflected in the line parallel to y-axis such that x = 4.

Solution

For any point P(a,b), its reflection P' in the x-axis will be P'(a,-b)
Since P'(a,-b) = P'(2,-3)
∴ Coordinates of P = P(a,b) = (2,3)
a = 2
b = 3

For any point P(a,b), its reflection P'' in the y-axis will be P''(-a,b)
Since P(a,b) = P(2,3)
∴ Coordinates of P'' = P''(-a,b) = (-2,3)

Graph to calculate P''':

Ml Aggarwal ICSE class 10 maths coordinate geometry reflection q13 Graph

Explanation:

Take 1cm = 1 unit on both axes.
Plot the line x = 4 that will cross x-axis at point (4,0) and will be parallel to y-axis.
Plot the point P(2,3), P'(2,-3) and P''(-2,3) on the graph.
From Graph: Reflection of P(2,3) on line x= 4 is P''' = (6,3)

Question 14

(i) Point P (a, b) is reflected in the x-axis to P’ (5, -2). Write down the values of a and b.
(ii) P” is the image of P when reflected in the y-axis. Write down the co-ordinates of P”.
(iii) Name a single transformation that maps P’ to P”.

Solution

(i) For any point P(a,b), its reflection P' in the x-axis will be P'(a,-b)
Since P'(a,-b) = P'(5,-2)
∴ Coordinates of P = P(a,b) = (5,2)
a = 5
b = 2

(ii) For any point P(a,b), its reflection P'' in the y-axis will be P''(-a,b)
Since P(a,b) = P(5,2)
∴ Coordinates of P'' = P''(-a,b) = (-5,2)

(iii) P'(5,-2) and P''(-5,2) maps by reflection in the origin

Question 15

Points A and B have co-ordinates (2, 5) and (0, 3). Find
(i) the image A’ of A under reflection in the x-axis.
(ii) the image B’ of B under reflection in the line AA’.

Solution

(i) For any point A(a,b), its reflection A' in the x-axis will be A'(a,-b)
Since A(a,b) = A'(2,5)
∴ Coordinates of A' = A'(a,-b) = (2,-5)

Question 16

Plot the points A (2, -3), B (-1, 2) and C (0, -2) on the graph paper. Draw the triangle formed by reflecting these points in the x-axis. Are the two triangles congruent?

Solution

Graph:

Explanation:

The points A (2, -3), B (-1, 2) and C(0, -2) has been plotted on the graph paper as shown and are joined to form a triangle ABC.
Hence, the co-ordinates of the images of A, B and C reflected in x-axis will be A’ (2, 3), B’ (-1, -2), C’ (0, 2) respectively.
And, these are joined to from another ∆ A’B’C’
Yes, these two triangles are congruent. (Measure the sides and their angles from the graph, they are identical).

Question 17

The points (6, 2), (3, -1) and (-2, 4) are the vertices of a right-angled triangle. Check whether it remains a right-angled triangle after reflection in the y-axis.

Solution

Graph:

Explanation:

Question 18

The triangle ABC where A (1, 2), B (4, 8), C (6, 8) is reflected in the x-axis to triangle A’ B’ C’. The triangle A’ B’ C’ is then reflected in the origin to triangle A”B”C” Write down the coordinates of A”, B”, C”. Write down a single transformation that maps ABC onto A” B” C”.

Solution

For any point P(a,b), its reflection P' in the x-axis will be P'(a,-b)
∴ Coordinates of A' = (1,-2)
∴ Coordinates of B' = (4,-8)
∴ Coordinates of C' = (6,-8)

For any point P(a,b), its reflection P' in the x-origin will be P'(-a,-b)

∴ Coordinates of image of A'(1,-2) in origin = A''(-1,2)
∴ Coordinates of image of B'(1,-2) in origin = B''(-4,8)
∴ Coordinates of image of C'(1,-2) in origin = B''(-6,8)

Comparing coordinates of △ABC and △A''B''C'', single transformation mapping points is reflection in y-axis.

Question 19

The image of a point P on reflection in a line l is point P’. Describe the location of the line l.

Solution

Since P and P' will be at equal distance from line l, we can say that the line will be the right bisector of the line segment joining P and P’.

Question 20

Given two points P and Q, and that (1) the image of P on reflection in the y-axis is the point Q and (2) the midpoint of PQ is invariant on reflection in x-axis. Locate: (i) the x-axis (ii) the y-axis and (iii) the origin.