![A triangle XYZ has vertices X(2, -1), Y(4, -1) and Z(4, 2). a) Find the coordinates of the image triangle X'Y'Z' after the transformation T1 = [[1, -3], [0, 1]]. b) Find the coordinates of the image t](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1777237593078-31ed1033ef0a5a41.png&w=3840&q=75)
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.
ScanSolve AI Answer
Welcome back Sewercydaa — missed you this week. Here is the solution to question 16: a) Step 1: Write the coordinates of triangle XYZ as column vectors. X = 2 \\ -1 , Y = 4 \\ -1 , Z = 4 \\ 2 Step 2: Apply transformation T_1 = 1 & -3 \\ 0 & 1 to find the coordinates of X'Y'Z'. X' = 1 & -3 \\ 0 & 1 2 \\ -1 = 1(2) + (-3)(-1) \\ 0(2) + 1(-1) = 2 + 3 \\ 0 - 1 = 5 \\ -1 Y' = 1 & -3 \\ 0 & 1 4 \\ -1 = 1(4) + (-3)(-1) \\ 0(4) + 1(-1) = 4 + 3 \\ 0 - 1 = 7 \\ -1 Z' = 1 & -3 \\ 0 & 1 4 \\ 2 = 1(4) + (-3)(2) \\ 0(4) + 1(2) = 4 - 6 \\ 0 + 2 = -2 \\ 2 The coordinates of triangle X'Y'Z' are X'(5, -1), Y'(7, -1), Z'(-2, 2). (Drawing on a grid is not possible in this format, but these are the coordinates you would plot). b) Step 1: Apply transformation T_2 = 0 & 1 \\ 1 & 0 to the coordinates of X'Y'Z' to find X''Y''Z''. X'' = 0 & 1 \\ 1 & 0 5 \\ -1 = 0(5) + 1(-1) \\ 1(5) + 0(-1) = -1 \\ 5 Y'' = 0 & 1 \\ 1 & 0 7 \\ -1 = 0(7) + 1(-1) \\ 1(7) + 0(-1) = -1 \\ 7 Z'' = 0 & 1 \\ 1 & 0 -2 \\ 2 = 0(-2) + 1(2) \\ 1(-2) + 0(2) = 2 \\ -2 The coordinates of triangle X''Y''Z'' are X''(-1, 5), Y''(-1, 7), Z''(2, -2). (Drawing on a grid is not possible in this format, but these are the coordinates you would plot). c) Step 1: The single transformation matrix T that maps XYZ to X''Y''Z'' is the product of T_2 and T_1, in that order (T_2 first, then T_1 applied to the original object, so T = T_2 T_1). T = T_2 T_1 = 0 & 1 \\ 1 & 0 1 & -3 \\ 0 & 1 Step 2: Perform the matrix multiplication. T = (0)(1) + (1)(0) & (0)(-3) + (1)(1) \\ (1)(1) + (0)(0) & (1)(-3) + (0)(1) T = 0 + 0 & 0 + 1 \\ 1 + 0 & -3 + 0 T = 0 & 1 \\ 1 & -3 d) Step 1: The area of the image is given by the formula: Area of image = |(T)| × Area of object. Step 2: Calculate the determinant of the transformation matrix T. T = 0 & 1 \\ 1 & -3 (T) = (0)(-3) - (1)(1) = 0 - 1 = -1 Step 3: Calculate the area of triangle X''Y''Z''. Area(X''Y''Z'') = |(T)| × Area(XYZ) Area(X''Y''Z'') = |-1| × 15 cm^2 Area(X''Y''Z'') = 1 × 15 cm^2 Area(X''Y''Z'') = 15 cm^2 Send me the next one 📸