Identify the coordinates of a point from the original triangle and its corresponding image point.

Here are the solutions to your questions. 16. If M is a translation vector that maps triangle XYZ into X'Y'Z', find the vector M. Step 1: Identify the coordinates of a point from the original triangle and its corresponding image point. Let's use point X and its image X'. From the graph, the coordinates are: X = (3, 0) X' = (0, 3) Step 2: Calculate the translation vector M. A translation vector M = a \\ b maps a point (x, y) to (x+a, y+b). So, X' = X + M. To find M, we subtract the original point's coordinates from the image point's coordinates: M = X' - X M = 0 \\ 3 - 3 \\ 0 M = 0 - 3 \\ 3 - 0 M = -3 \\ 3 The translation vector is -3 \\ 3 . 17. A quantity P varies directly as q and inversely as the cube root of r. When p = 10 and q = 5 then r = 8. Find P when q = 54 and r = 27. Step 1: Write the variation equation. "P varies directly as q" means P q. "P varies inversely as the cube root of r" means P (1)/([3]r). Combining these, we get the relationship: P = (kq)/([3]r) where k is the constant of proportionality. Step 2: Use the given values to find the constant of proportionality, k. Given P = 10, q = 5, and r = 8. Substitute these values into the equation: 10 = (k × 5)/([3]8) Calculate the cube root of 8: [3]8 = 2. 10 = (5k)/(2) Multiply both sides by 2: 20 = 5k Divide by 5: k = (20)/(5) k = 4 Step 3: Write the complete variation equation. Substitute the value of k back into the equation: P = (4q)/([3]r) Step 4: Find P when q = 54 and r = 27. Substitute q = 54 and r = 27 into the complete variation equation: P = (4 × 54)/([3]27) Calculate the cube root of 27: [3]27 = 3. P = (4 × 54)/(3) P = (216)/(3) P = 72 The value of P is 72. That's 4 down. 1 left today — send the next one.