Mathematics
Points (3, 0) and (-1, 0) are invariant points under reflection in the line L1; points (0, -3) and (0, 1) are invariant points on reflection in line L2.
(i) Name and write equations for the lines L1 and L2.
(ii) Write down the images of points P(3, 4) and Q(-5, -2) on reflection in L1. Name the images as P' and Q' respectively.
(iii) Write down the images of P and Q on reflection in L2. Name the images as P" and Q" respectively.
(iv) State or describe a single transformation that maps P' onto P".
Answer
(i) We know that every point in a line is invariant under the reflection in the same line.
Since, the points (3, 0) and (-1, 0) lie on the x-axis.
So, points (3, 0) and (-1, 0) are invariant under reflection in x-axis.
So, L1 = x axis.
Since, the points (0, -3) and (0, 1) lie on the y-axis.
So, points (0, -3) and (0, 1) are invariant under reflection in y-axis.
So, L2 = y axis.
Hence, L1 = x-axis whose equation is y = 0 and L2 = y-axis whose equation is x = 0.
(ii) Line L1 is x axis.
Reflection in x-axis is given by,
Mx(x, y) = (x, -y)
∴ Image on reflection of P(3, 4) in L1 (x-axis) = P'(3, -4)
Similarly, image on reflection of Q(-5, -2) in L1 (x-axis) = Q'(-5, 2)
Hence, co-ordinates of P' = (3, -4) and Q' = (-5, 2).
(iii) Line L2 is y axis.
Reflection in y-axis is given by,
My(x, y) = (-x, y)
∴ Image on reflection of P(3, 4) in L2 (y-axis) = P''(-3, 4)
Similarly, image on reflection of Q(-5, -2) in L2 (y-axis) = Q''(5, -2)
Hence, co-ordinates of P" = (-3, 4) and Q" = (5, -2).
(iv) P' = (3, -4) and P" = (-3, 4)
P'(3, -4) ⇒ P"(-3, 4)
Since sign of both abscissa and ordinate is changed, this transformation is possible on reflection in origin.
Hence, reflection in origin maps P' onto P".
Related Questions
The triangle ABC, where A is (2, 6), B is (-3, 5) and C is (4, 7), is reflected in the y-axis to triangle A'B'C'. Triangle A'B'C' is then reflected in the origin to triangle A"B"C".
(i) Write down the co-ordinates of A", B" and C".
(ii) Write down a single transformation that maps triangle ABC onto triangle A"B"C".
Attempt this question on graph paper.
(a) Plot A (3, 2) and B (5, 4) on graph paper. Take 2 cm = 1 unit on both the axes.
(b) Reflect A and B in the x-axis to A' and B' respectively. Plot these points also on the same graph paper.
(c) Write down :
(i) the geometrical name of the figure ABB'A';
(ii) the measure of angle ABB';
(iii) the image A" of A, when A is reflected in the origin.
(iv) the single transformation that maps A' to A".
The point (-2, 0) on reflection in a line is mapped to (2, 0) and the point (5, -6) on reflection in the same line is mapped to (-5, -6).
(i) State the name of mirror line and write its equation.
(ii) State the co-ordinates of the image of (-8, -5) in the mirror line.
The points P(4, 1) and Q(-2, 4) are reflected in line y = 3. Find the co-ordinates of P', the image of P and Q', the image of Q.