**4. A triangle XYZ with vertices X(1, 2), Y(-1, -2) and Z(5, 0) is translated by a vector 1 2 . The image so formed is reflected on the x-axis. Write the co-ordinate
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4. A triangle XYZ with vertices X(1, 2), Y(-1, -2) and Z(5, 0) is translated by a vector $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$. The image so formed is reflected on the x-axis. Write the co-ordinates of the vertices of the images thus obtained and represent the $\triangle XYZ$ and its images on the same graph paper.
Step 1: Translation
The translation vector is $T = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$.
The formula for translation is $P'(x', y') = P(x, y) + T$.
For X(1, 2):
$$X' = (1+1, 2+2) = (2, 4)$$
For Y(-1, -2):
$$Y' = (-1+1, -2+2) = (0, 0)$$
For Z(5, 0):
$$Z' = (5+1, 0+2) = (6, 2)$$
The coordinates after translation are $\boxed{X'(2, 4), Y'(0, 0), Z'(6, 2)}$.
Step 2: Reflection on the x-axis
The formula for reflection on the x-axis is $(x, y) \to (x, -y)$.
For X'(2, 4):
$$X'' = (2, -4)$$
For Y'(0, 0):
$$Y'' = (0, 0)$$
For Z'(6, 2):
$$Z'' = (6, -2)$$
The coordinates after reflection are $\boxed{X''(2, -4), Y''(0, 0), Z''(6, -2)}$.
Step 3: Graph Representation
Plot $\triangle XYZ$ with vertices X(1, 2), Y(-1, -2), Z(5, 0).
Plot $\triangle X'Y'Z'$ with vertices X'(2, 4), Y'(0, 0), Z'(6, 2).
Plot $\triangle X''Y''Z''$ with vertices X''(2, -4), Y''(0, 0), Z''(6, -2).
This step should be performed on graph paper.
5. $\triangle PQR$ with vertices P(2, 1), Q(2, 4) and R(5, 2) is translated by the vector $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$ and then the image so formed is rotated about the origin through $90^\circ$ in positive direction. Write down the co-ordinates of the vertices of the images thus obtained and present $\triangle PQR$ and its images in the same graph paper.
Step 1: Translation
The translation vector is $T = \begin{pmatrix} 1 \\ -2 \end{pmatrix}$.
For P(2, 1):
$$P' = (2+1, 1-2) = (3, -1)$$
For Q(2, 4):
$$Q' = (2+1, 4-2) = (3, 2)$$
For R(5, 2):
$$R' = (5+1, 2-2) = (6, 0)$$
The coordinates after translation are $\boxed{P'(3, -1), Q'(3, 2), R'(6, 0)}$.
Step 2: Rotation about the origin through $90^\circ$ in positive direction (counter-clockwise)
The formula for rotation about the origin $90^\circ$ counter-clockwise is $(x, y) \to (-y, x)$.
For P'(3, -1):
$$P'' = (-(-1), 3) = (1, 3)$$
For Q'(3, 2):
$$Q'' = (-2, 3)$$
For R'(6, 0):
$$R'' = (-0, 6) = (0, 6)$$
The coordinates after rotation are $\boxed{P''(1, 3), Q''(-2, 3), R''(0, 6)}$.
Step 3: Graph Representation
Plot $\triangle PQR$ with vertices P(2, 1), Q(2, 4), R(5, 2).
Plot $\triangle P'Q'R'$ with vertices P'(3, -1), Q'(3, 2), R'(6, 0).
Plot $\triangle P''Q''R''$ with vertices P''(1, 3), Q''(-2, 3), R''(0, 6).
This step should be performed on graph paper.
**6. A triangle LMN with vertices L(-3, 3), M(-6, 3) and N(-3, 9) respectively is reflected in y-axis and then image formed is rotated about the origin through $180^\circ$. Write down the co-ordinates of the
