# Frank Solutions for Chapter 8 Reflection Class 10 ICSE Mathematics

**Exercise 8**

**1. State the co-ordinates of the images of the following points under reflection in the x-axis:**

(i) (3,-9)

**(ii) (-7, 5)**

**(iii) (0, 6)**

**(iv) (-4,-8)**

**Answer**

**(i)** (3,-9)

The co-ordinates of the images of the given points under reflection in the x-axis is (3, 9)

**(ii)** (-7, 5)

The co-ordinates of the images of the given points under reflection in the x-axis is (-7, 5)

**(iii)** (0, 6)

The co-ordinates of the images of the given points under reflection in the x-axis is (0, 6)

**(iv)** (-4,-8)

The co-ordinates of the images of the given points under reflection in the x-axis is (-4, 8)

**2. ****State the co-ordinates of the images of the following points under reflection in the y-axis:**

**(i) (2, 8)**

**(ii) (-1,-3)**

**(iii) (5,-6)**

**(iv) (-4, 7)**

**Answer**

**(i)** (2, 8)

The co-ordinates of the images of the given points under reflection in the y-axis is (-2, 8)

**(ii)** (-1,-3)

The co-ordinates of the images of the given points under reflection in the y-axis is (1, -3)

**(iii)** (5,-6)

The co-ordinates of the images of the given points under reflection in the y-axis is (-5, -6)

**(iv)** (-4, 7)

The co-ordinates of the images of the given points under reflection in the y-axis is (4, 7)

**3. State the co-ordinates of the images of the following points under reflection in the origin:**

**(i) (-1,-4)**

**(ii) (2, 7)**

**(iii) (0, 2)**

**(iv) (9,-9)**

**Answer**

**(i)** (-1,-4)

The co-ordinates of the images of the given points under reflection in the origin is (1, 4)

**(ii)** (2, 7)

The co-ordinates of the images of the given points under reflection in the origin is (-2, -7)

**(iii)** (0, 2)

The co-ordinates of the images of the given points under reflection in the origin is (0, -2)

**(iv)** (9,-9)

The co-ordinates of the images of the given points under reflection in the origin is (-9, 9)

**4. ****P’ is the image of P under reflection in the x-axis. If the co-ordinates of P’ are (2, 10), write the co-ordinates of P.**

**Answer**

From the question it is given that, P’ = (2, 10)

Then, the co-ordinates of P under reflection in the x-axis is (2, -10)

**5.** **S’ is the image of S under reflection in the origin. If the co-ordinates of S are (2,-5), write the co-ordinates of S’.**

**Answer**

From the question it is given that, S = (2, -5)

Then, the co-ordinates of S’ under reflection in the origin is (-2, 5)

**6. ****A point P is reflected in the x-axis to P’. P’ is then reflected in the origin to P”. If the co-ordinates of P’ are (-3, 4). Find the co-ordinates of P and P”. Write the single transformation that map P onto P”.**

**Answer**

From the question it is given that,

P’ = (-3, 4)

Then, the co-ordinates of P under reflection in the x-axis is (-3, -4)

Also the co-ordinates of P” under reflection in the origin is (3, -4)

The single transformation = reflection in the y-axis.

**7. ****A point P is reflected in the x-axis. Co-ordinates of its image are (8, -6). Find the co-ordinates of P. Find the co-ordinates of the image of P under reflection in the y-axis.**

**Answer**

From the question it is given that,

P’ = (8, -6)

Then, the co-ordinates of P under reflection in the x -axis is (8, 6)

The co-ordinates of P” under reflection in the y – axis is (-8, 6)

**8. A point R (3, -2) is reflected in the origin as R’. Point Q (-7, 1) is reflected in the x-axis as Q’. Write down the co-ordinates of R’ and Q’. Calculate the distance R’ Q’.**

**Answer**

From the question it is given that

A point R = (3, -2)

Then, a point R is reflected in the origin as R’ = (-3, 2)

Point Q = (-7, 1)

Then, a point Q is reflected in the x – axis as Q’ = (-7, -1)

Now, we have to calculate the distance between R’ Q’ = √[(-7 – (-3))^{2} + (-1 – 2)^{2}]

= √[(-4)^{2} + (-3)^{2}]

= √[16 + 9]

= √25

= 5 units

**9. ****The points B and C have the co-ordinates (3, 2) and (0, 3). Find B’, the image of B under the reflection in the x-axis and C’, the image of C under the reflection in the line BB’.**

**Answer**

From the question it is given that,

Point B = (3, 2)

Point C = (0, 3)

Then, the reflection of B in the x – axis is B’ = (3, -2)

And the reflection of C in the line BB’ is C’ = (6, 3)

**10.** **A point P is mapped onto P’ under the reflection in the x-axis. P’ is mapped onto P” under the reflection in the origin. If the co-ordinates of ****P” are (5,-2), write down the co-ordinates of P. State the single transformation that takes place.**

**Answer**

From the question it is given that,

P” = (5, -2)

Then, co-ordinates of P’ = (-5, 2)

Therefore, the co-ordinates of P = (-5, -2)

Single transformation = reflection in the y-axis

**11. ****Write down the co-ordinates of the image of the point (-2, 4) under reflection in the origin and under reflection in the y-axis. What is the distance between the points of reflection?**

**Answer**

Let us assume that S be the Point,

So, S = (-2, 4)

Then, image under reflection in the origin S’ = (2, -4)

And image under reflection in the y-axis S” = (2, 4)

Now, the distance between point of reflection = √[(4 – (-4))^{2} + (2 – 2)^{2}]

= √[(8)^{2} + (0)^{2}]

= √64

= 8 units

**12.** **A triangle ABC lies in the co – ordinate plane. The co – ordinates of its vertices are A (2, 3), B (4, -4) and C (6, -7). This triangle is reflected in the line y = 0 on to Î”A’B’C’. Î”A’B’C’ in then reflected in the origin onto Î”A”B”C”. Write down the co-ordinates of Î”A’B’C’ and Î”A”B”C”.**

**Answer**

From the question it is given that,

The co – ordinates of its vertices are A (2, 3), B (4, -4) and C (6, -7)

Then, co-ordinates of Î”A’B’C’ under reflection in the line y = 0.

So, A’ = (2, -3)

B’ = (4, 4)

C’ = (6, 7)

Now, co-ordinates of Î”A”B”C” under reflection in the origin,

A” = (-2, 3)

B” = (-4, -4)

C” = (-6, -7)

**13.** **A point P (-8, 1) is reflected in the x-axis to the point P’. The point P’ is then reflected in the origin to point P”. Write down the co-ordinates of P”. State the single transformation that maps P into P”.**

**Answer**

From the question it is given that, P = (-8, 1)

The co-ordinates of P’ under reflection in the x –axis = (-8, -1)

Then, the co-ordinates of p” under reflection in the origin = (8, 1)

Single transformation = reflection in the y-axis

**14. Perform the following operations and state the single transformation that takes place in each.**

**(i) M _{x}.M_{y} on P (2, -5)**

**(ii) M _{y}.M_{o} on A (-7, 3)**

**(iii) M _{o}.M_{y} on B (4, 6)**

**(iv) M _{x}.M_{o} on P (-1, -3)**

**Answer**

**(i)** M_{x}.M_{y} on P (2, -5)

M_{x}.M_{y} (2, -5)

M_{x} (-2, -5)

Therefore, reflection in the origin is (-2, 5)

**(ii)** M_{y}.M_{o} on A (-7, 3)

M_{y}.M_{o} (-7, 3)

M_{y} (7, -3)

Therefore, reflection in the x- axis is (-7, -3)

**(iii)** M_{o}.M_{y} on B (4, 6)

M_{o}.M_{y} (4, 6)

M_{o} (-4, 6)

Therefore, reflection in the x – axis (4, -6)

**(iv)** M_{x}.M_{o} on P (-1, -3)

M_{x}.M_{o} (-1, -3)

M_{x} (-1, 3)

Therefore, reflection in the y – axis (-1, -3)

**15. ****Find the co – ordinates of the image of A (-5, 4) after reflection in the line.**

**(i) y = 0**

**(i) y = 4**

**Answer**

**(i)** y = 0

Co – ordinates of the image = (-5, (2 × 0) – 4)

= (-5, (0 – 4))

= (-5, -4)

**(ii)** y = 4

Co – ordinates of the image = (-5, (2 × 4) – 4)

= (-5, (8 – 4))

= (-5, 4)

**16. Find the co-ordinates of the image of S (4, - 1) after reflection in the line **

**(i) x = 0 **

**(ii) y = 5 **

**Answer**

**(i)** x = 0

Co-ordinates of image = (2 × 0 – 4, 1) = (-4, -1)

**(ii)** y = 5

Co-ordinates of image = (4, 2 × 5 – (-1) = (4, 11)

**17. The point P (1, - 1) , Q (4, -1) and R( 4, 3) are reflected in y-axis. Ig=f the images are denoted by P’, Q’ and R’, then **

**(i) Find the co-ordinates of P’Q’R, Q’ and R’ .**

**(ii) What kind of figures is formed by RR’Q’Q ? **

**(iii) Find the perimeter of the figure P’Q’R **

**Answer**

**(i)** Find the co-ordinates of P’Q’R’, Q’ and R’

**18. Point A (1, - 5) is mapped as A’ on reflection in the line y = 1. The point B (- 5, 1) mapped as B’ on reflection in the line y = 4. Write the co-ordinates of A’ and B’. Calculate AB’.**

**Answer**

A (1, -5), the co-ordinates of A’ (1, 2×1 – (-5)) = (1, 7)

B (- 5, 1), the co-ordinates of B’ = (- 5, 2×4 – (1) = (- 5, 7)

**19. Point A (4, - 1) is reflected as A’ in the line x = 1. Point B on reflection in the line y = 3 is mapped as B’ (6, - 1). Write the co-ordinates of A’ and B. Write the co-ordinates of mid-point of the line segment A’B’.**

**Answer**

A (4, -1), the co-ordinates of A’ (2 × 1 – 4, - 1) = (- 2, - 1)