**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

Here are the solutions to the transformation problems. 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-ordinates of the vertices of the images thus obtained and represent the XYZ and its images on the same graph paper. Step 1: Translation The translation vector is T = 1 \\ 2 . 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 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) (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 X''(2, -4), Y''(0, 0), Z''(6, -2). Step 3: Graph Representation Plot XYZ with vertices X(1, 2), Y(-1, -2), Z(5, 0). Plot X'Y'Z' with vertices X'(2, 4), Y'(0, 0), Z'(6, 2). Plot X''Y''Z'' with vertices X''(2, -4), Y''(0, 0), Z''(6, -2). This step should be performed on graph paper. 5. PQR with vertices P(2, 1), Q(2, 4) and R(5, 2) is translated by the vector 1 \\ -2 and then the image so formed is rotated about the origin through 90^ in positive direction. Write down the co-ordinates of the vertices of the images thus obtained and present PQR and its images in the same graph paper. Step 1: Translation The translation vector is T = 1 \\ -2 . 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 P'(3, -1), Q'(3, 2), R'(6, 0). Step 2: Rotation about the origin through 90^ in positive direction (counter-clockwise) The formula for rotation about the origin 90^ counter-clockwise is (x, y) (-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 P''(1, 3), Q''(-2, 3), R''(0, 6). Step 3: Graph Representation Plot PQR with vertices P(2, 1), Q(2, 4), R(5, 2). Plot P'Q'R' with vertices P'(3, -1), Q'(3, 2), R'(6, 0). Plot 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^. Write down the co-ordinates of the