# Reflection

- The reflection (or image) of a point P in a line AB is a point P'such that AB is the perpendicular bisector of the line segment PP'.
- The reflection (or image) of a point P in a given point M is a point P' such that M is the mid-point of the line segment PP'.
*Reflection*

(i) Reflection of P(x, y) in the x-axis is P'(x, -y).

(ii) Reflection of P(x, y) in the y-axis is P'(-x, y).

(iii) Reflection of P(x, y) in the origin is P'(-x, -y).

## Exercise

- The image of a point P under reflection in the x-axis is (-3, 5), find the co-ordinates of P.
- A point P is reflected in the x-axis. Co-ordinates of its image are (8, -6). Find

(i) the co-ordinates of P.

(ii) the co-ordinates of the image of P under reflection in the y-axis. - The point P(4, -7) on reflection in x-axis is mapped onto P'. Then P' on reflection in the y-axis is mapped onto P''. Find the co-ordinates of P' and P''. Write down a single transformation that maps P onto P''.
- The point P(a, b) is first reflected in the origin and then reflected in the x-axis to P' has co-ordinates (-3, 5), find the values of a and b.
- Find p and q if

(i) (-2, 3) on reflection in x-axis is mapped at (p, q).

(ii) (3, p) on reflection in y-axis is mapped at (q -1, 4).

(iii) (p, 3) on reflection in the origin is mapped at (-2, -q +5). - 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'. - Plot the points A(1, -2), B(-2, 5) and C(-3, -4) on the graph paper. Draw the triangle formed by reflecting these points in the y-axis. Are the two triangles congruent?
- 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 co-ordinates of A'', B'', C''. Write down a single transformation that maps ABC to A''B''C''.
- The points A(2, -3), B(3, 4) and C(7, 5) are the vertices of
ABC.

(i) Write the co-ordinates of A_{1}, B_{1}, C_{1}where ABC is reflected in the x-axis to A_{1}B_{1}C_{1}.

(ii) Write the co-ordinates of A_{2}, B_{2}, C_{2}when A_{1}B_{1}C_{1}is reflected in the y-axis to A_{2}B_{2}C_{2}.

(iii) Write the co-ordinates of A_{3}, B_{3}, C_{3}when A_{2}B_{2}C_{2}is reflected in the origin to A_{3}B_{3}C_{3}.

(iv) What conclusion can you derive from the above reflections? - The points A(0, -1), B(-2, 3), C(6, 7) and D(8, 3) are the vertices of a rectangle. If the rectangle, ABCD is reflected in the y-axis and then in the origin, find the co-ordinates of the final images. Check whether it remains a rectangle. Write down a single transformation that brings the above change.
- 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 the line L_{2}.

(i) Name the lines L_{1}and L_{2}.

(ii) Write down the images of points P(3, 4) and Q(-5, -2) on reflection in 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''. - Attempt this question on graph paper.

(i) Plot A(3, 2) and B(5, 4) on the graph paper. Take 2 cm = 1 unit on both axes.

(ii) Reflect A and B in the x-axis to A', B'. Plot these on the same graph paper.

(iii) Write down

(a) the geometrical name of the figure ABB'A'.

(b) the axis of symmetry of ABB'A'.

(c) the measure of the angle ABB'.

(d) the image A'' of A, when A is reflected in the origin.

(e) the single transformation that maps A' to A''.

## Answers

**1.**(-3, -5)

**2.**(i) (8, 6) (ii) (-8, 6)

**3.**P'(4, 7), P''(-4, 7); reflection in the origin.

**4.**a = 3, b = 5

**5.**(i) p = -2, q = -3 (ii) p = 4, q = -2 (iii) p = 2, q = 8

**6.**(i) (2, -5) (ii) (4, 3)

**7.**Yes

**8.**A''(-1, 2), B''(-4, 8), C''(-6, 8); reflection in the y-axis.

**9.**(i) A

_{1}(2, 3), B

_{1}(3, -4), C

_{1}(7, -5)

(ii) A

_{2}(-2, 3), B

_{2}(-3, -4), C

_{2}(-7, -5)

(iii) A

_{3}(2, -3), B

_{3}(3, 4), C

_{3}(7, 5)

(iv) A

_{3}B

_{3}C

_{3}coincides with ABC.

**10.**(0, 1), (-2, -3), (6, -7) and (8, -3); yes, reflection in the x-axis.

**11.**(i) L

_{1}-x-axis; L

_{2}-y-axis

(ii) P'(3, -4), Q'(-5, 2)

(iii) P''(-3, 4), Q''(5, -2)

(iv) reflection through origin.

**12.**(ii) A'(3, -2), B'(5, -4)

(iii) (a) Isosceles trapezium (b) x-axis

(c) 90° (d) (-3, -2) (e) reflection in y-axis