This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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a) I cannot draw the graph for you, but you should draw two perpendicular axes, Ox and Oy, on graph paper. The x-axis should range from -10 to 10, and the y-axis from -12 to 12. Use a scale where 2 cm represents 2 units on both axes. b) I cannot draw the triangles, but I will provide the coordinates for each vertex. i) The vertices of triangle ABC are given as: A(4, 4) B(6, 8) C(2, 8) ii) To find the image A_1B_1C_1 of ABC under a reflection in the line x = -2, we use the transformation rule (x, y) (2k - x, y), where k = -2. So, (x, y) (-4 - x, y). Step 1: Apply the reflection rule to point A. A(4, 4) A_1(-4 - 4, 4) = A_1(-8, 4) Step 2: Apply the reflection rule to point B. B(6, 8) B_1(-4 - 6, 8) = B_1(-10, 8) Step 3: Apply the reflection rule to point C. C(2, 8) C_1(-4 - 2, 8) = C_1(-6, 8) The vertices of A_1B_1C_1 are A_1(-8, 4), B_1(-10, 8), C_1(-6, 8). iii) To find the image A_2B_2C_2 of ABC under an enlargement with scale factor -1 from the origin, we use the transformation rule (x, y) (kx, ky), where k = -1. So, (x, y) (-x, -y). Step 1: Apply the enlargement rule to point A. A(4, 4) A_2(-4, -4) Step 2: Apply the enlargement rule to point B. B(6, 8) B_2(-6, -8) Step 3: Apply the enlargement rule to point C. C(2, 8) C_2(-2, -8) The vertices of A_2B_2C_2 are A_2(-4, -4), B_2(-6, -8), C_2(-2, -8). iv) To find the image A_3B_3C_3 of ABC under a reflection in the line y = 0 (the x-axis), we use the transformation rule (x, y) (x, 2k - y), where k = 0. So, (x, y) (x, -y). Step 1: Apply the reflection rule to point A. A(4, 4) A_3(4, -4) Step 2: Apply the reflection rule to point B. B(6, 8) B_3(6, -8) Step 3: Apply the reflection rule to point C. C(2, 8) C_3(2, -8) The vertices of A_3B_3C_3 are A_3(4, -4), B_3(6, -8), C_3(2, -8). c) To find the single transformation that maps A_2B_2C_2 onto A_3B_3C_3, we compare their coordinates: A_2(-4, -4) maps to A_3(4, -4) B_2(-6, -8) maps to B_3(6, -8) C_2(-2, -8) maps to C_3(2, -8) Step 1: Observe the change in coordinates from A_2 to A_3. The y-coordinate remains the same (e.g., -4 -4), while the x-coordinate changes sign (e.g., -4 4). Step 2: Confirm this pattern for B_2 B_3 and C_2 C_3. For B: (-6, -8) (6, -8). x -x, y y. For C: (-2, -8) (2, -8). x -x, y y. Step 3: Identify the transformation. The transformation rule is (x, y) (-x, y). This is a reflection in the y-axis (the line x=0). The single transformation is a reflection in the y-axis. Send me the next one 📸