Triangle PQR has vertices P(2,6), Q(2,2), and R(4,6). Find the image of triangle PQR under the transformations given by matrices a) [[0, 1], [1, 0]] and b) [[1, 2], [-1, 1]].

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Asked on April 8, 2026Mathematics

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.

Accepted Answer · 2026-04-08T06:36:53.563Z

Here are the images of triangle PQR under the given transformations. The vertices of triangle PQR are P(2,6), Q(2,2), and R(4,6). a) For the matrix 0 & 1 \\ 1 & 0 : Step 1: Calculate the image of point P. P' = 0 & 1 \\ 1 & 0 2 \\ 6 = (0)(2) + (1)(6) \\ (1)(2) + (0)(6) = 6 \\ 2 Step 2: Calculate the image of point Q. Q' = 0 & 1 \\ 1 & 0 2 \\ 2 = (0)(2) + (1)(2) \\ (1)(2) + (0)(2) = 2 \\ 2 Step 3: Calculate the image of point R. R' = 0 & 1 \\ 1 & 0 4 \\ 6 = (0)(4) + (1)(6) \\ (1)(4) + (0)(6) = 6 \\ 4 The image of triangle PQR is P'(6,2), Q'(2,2), R'(6,4). b) For the matrix 1 & 2 \\ -1 & 1 : Step 1: Calculate the image of point P. P' = 1 & 2 \\ -1 & 1 2 \\ 6 = (1)(2) + (2)(6) \\ (-1)(2) + (1)(6) = 2 + 12 \\ -2 + 6 = 14 \\ 4 Step 2: Calculate the image of point Q. Q' = 1 & 2 \\ -1 & 1 2 \\ 2 = (1)(2) + (2)(2) \\ (-1)(2) + (1)(2) = 2 + 4 \\ -2 + 2 = 6 \\ 0 Step 3: Calculate the image of point R. R' = 1 & 2 \\ -1 & 1 4 \\ 6 = (1)(4) + (2)(6) \\ (-1)(4) + (1)(6) = 4 + 12 \\ -4 + 6 = 16 \\ 2 The image of triangle PQR is P'(14,4), Q'(6,0), R'(16,2). c) For the matrix -2 & 0 \\ 0 & -3 : Step 1: Calculate the image of point P. P' = -2 & 0 \\ 0 & -3 2 \\ 6 = (-2)(2) + (0)(6) \\ (0)(2) + (-3)(6) = -4 \\ -18 Step 2: Calculate the image of point Q. Q' = -2 & 0 \\ 0 & -3 2 \\ 2 = (-2)(2) + (0)(2) \\ (0)(2) + (-3)(2) = -4 \\ -6 Step 3: Calculate the image of point R. R' = -2 & 0 \\ 0 & -3 4 \\ 6 = (-2)(4) + (0)(6) \\ (0)(4) + (-3)(6) = -8 \\ -18 The image of triangle PQR is P'(-4,-18), Q'(-4,-6), R'(-8,-18). d) For the matrix (1)/(2) & (1)/(4) \\ (1)/(2) & (1)/(2) : Step 1: Calculate the image of point P. P' = (1)/(2) & (1)/(4) \\ (1)/(2) & (1)/(2) 2 \\ 6 = ((1)/(2))(2) + ((1)/(4))(6) \\ ((1)/(2))(2) + ((1)/(2))(6) = 1 + (6)/(4) \\ 1 + 3 = 1 + (3)/(2) \\ 4 = (2+3)/(2) \\ 4 = (5)/(2) \\ 4 Step 2: Calculate the image of point Q. Q' = (1)/(2) & (1)/(4) \\ (1)/(2) & (1)/(2) 2 \\ 2 = ((1)/(2))(2) + ((1)/(4))(2) \\ ((1)/(2))(2) + ((1)/(2))(2) = 1 + (2)/(4) \\ 1 + 1 = 1 + (1)/(2) \\ 2 = (2+1)/(2) \\ 2 = (3)/(2) \\ 2 Step 3: Calculate the image of point R. R' = (1)/(2) & (1)/(4) \\ (1)/(2) & (1)/(2) 4 \\ 6

Accepted Answer · 2026-04-08T06:36:53.563Z

Here are the images of triangle PQR under the given transformations. The vertices of triangle PQR are P(2,6), Q(2,2), and R(4,6). a) For the matrix 0 & 1 \\ 1 & 0 : Step 1: Calculate the image of point P. P' = 0 & 1 \\ 1 & 0 2 \\ 6 = (0)(2) + (1)(6) \\ (1)(2) + (0)(6) = 6 \\ 2 Step 2: Calculate the image of point Q. Q' = 0 & 1 \\ 1 & 0 2 \\ 2 = (0)(2) + (1)(2) \\ (1)(2) + (0)(2) = 2 \\ 2 Step 3: Calculate the image of point R. R' = 0 & 1 \\ 1 & 0 4 \\ 6 = (0)(4) + (1)(6) \\ (1)(4) + (0)(6) = 6 \\ 4 The image of triangle PQR is P'(6,2), Q'(2,2), R'(6,4). b) For the matrix 1 & 2 \\ -1 & 1 : Step 1: Calculate the image of point P. P' = 1 & 2 \\ -1 & 1 2 \\ 6 = (1)(2) + (2)(6) \\ (-1)(2) + (1)(6) = 2 + 12 \\ -2 + 6 = 14 \\ 4 Step 2: Calculate the image of point Q. Q' = 1 & 2 \\ -1 & 1 2 \\ 2 = (1)(2) + (2)(2) \\ (-1)(2) + (1)(2) = 2 + 4 \\ -2 + 2 = 6 \\ 0 Step 3: Calculate the image of point R. R' = 1 & 2 \\ -1 & 1 4 \\ 6 = (1)(4) + (2)(6) \\ (-1)(4) + (1)(6) = 4 + 12 \\ -4 + 6 = 16 \\ 2 The image of triangle PQR is P'(14,4), Q'(6,0), R'(16,2). c) For the matrix -2 & 0 \\ 0 & -3 : Step 1: Calculate the image of point P. P' = -2 & 0 \\ 0 & -3 2 \\ 6 = (-2)(2) + (0)(6) \\ (0)(2) + (-3)(6) = -4 \\ -18 Step 2: Calculate the image of point Q. Q' = -2 & 0 \\ 0 & -3 2 \\ 2 = (-2)(2) + (0)(2) \\ (0)(2) + (-3)(2) = -4 \\ -6 Step 3: Calculate the image of point R. R' = -2 & 0 \\ 0 & -3 4 \\ 6 = (-2)(4) + (0)(6) \\ (0)(4) + (-3)(6) = -8 \\ -18 The image of triangle PQR is P'(-4,-18), Q'(-4,-6), R'(-8,-18). d) For the matrix (1)/(2) & (1)/(4) \\ (1)/(2) & (1)/(2) : Step 1: Calculate the image of point P. P' = (1)/(2) & (1)/(4) \\ (1)/(2) & (1)/(2) 2 \\ 6 = ((1)/(2))(2) + ((1)/(4))(6) \\ ((1)/(2))(2) + ((1)/(2))(6) = 1 + (6)/(4) \\ 1 + 3 = 1 + (3)/(2) \\ 4 = (2+3)/(2) \\ 4 = (5)/(2) \\ 4 Step 2: Calculate the image of point Q. Q' = (1)/(2) & (1)/(4) \\ (1)/(2) & (1)/(2) 2 \\ 2 = ((1)/(2))(2) + ((1)/(4))(2) \\ ((1)/(2))(2) + ((1)/(2))(2) = 1 + (2)/(4) \\ 1 + 1 = 1 + (1)/(2) \\ 2 = (2+1)/(2) \\ 2 = (3)/(2) \\ 2 Step 3: Calculate the image of point R. R' = (1)/(2) & (1)/(4) \\ (1)/(2) & (1)/(2) 4 \\ 6

Triangle PQR has vertices P(2,6), Q(2,2), and R(4,6). Find the image of triangle PQR under the transformations given by matrices a) [[0, 1], [1, 0]] and b) [[1, 2], [-1, 1]].

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Triangle PQR has vertices P(2,6), Q(2,2), and R(4,6). Find the image of triangle PQR under the transformations given by matrices a) [[0, 1], [1, 0]] and b) [[1, 2], [-1, 1]].

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