Question 8 • Advanced Algebra: Concepts and Connections
A triangle is located on a three-dimensional coordinate plane. Matrix cap A shows the location of each vertex, where the first column represents the x -coordinate, the second column represents the y -coordinate, and the third column represents the z -coordinate. cap A is equal to the 3 by 3 matrix, row 1, column 1, 1 column 2, 2 column 3, 1 row 2, column 1, 3 column 2, 1 column 3, 0 row 3, column 1, 2 column 2, 4 column 3, negative 1 The triangle is dilated by a scale factor of 2 , then translated by the factors in matrix cap b , resulting in an image triangle as represented by the equation. 2 cap A plus cap b is equal to the 3 by 3 matrix, row 1, column 1, 5 column 2, 2 column 3, 3 row 2, column 1, 9 column 2, 0 column 3, 1 row 3, column 1, 7 column 2, 6 column 3, negative 1 In which directions are the vertices translated, as indicated by matrix cap b ?
Answer
A
3 units in the x direction, 3 units in the y direction, 3 units in the z direction
B
4 units in the x direction, 0 units in the y direction, 2 units in the z direction
C
4 units in the x direction, 6 units in the y direction, 5 units in the z direction
D
3 units in the x direction, negative 2 units in the y direction, 1 unit in the z direction
Explanation
Doubling each row of matrix A gives the dilated vertices: (2, 4, 2), (6, 2, 0), and (4, 8, −2). Subtract these coordinates from the corresponding image vertices; each difference is (3, −2, 1). So the translation is 3 units in the x direction, −2 units in the y direction, and 1 unit in the z direction, which is choice A.
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