Question
What is the final position of point A(7,8) after the a translation of two units left one unit down and followed by 180 degrees clockwise rotation around the origin
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GPT 3.5
To find the final position of point A(7,8) after a translation of two units left and one unit down, we subtract 2 from the x-coordinate and subtract 1 from the y-coordinate:
Translation: A(7-2, 8-1) = A(5, 7)
Next, to perform a 180 degrees clockwise rotation around the origin, we multiply the coordinates of point A by the matrix:
[cos(180), -sin(180)]
[sin(180), cos(180)]
The cosine of 180 degrees is -1, and the sine of 180 degrees is 0. Applying the matrix multiplication:
A'(x, y) = [5(-1) + 7(0), 5(0) + 7(-1)]
= [-5, -7]
Therefore, the final position of point A after the translation and rotation is A'(-5, -7).
Translation: A(7-2, 8-1) = A(5, 7)
Next, to perform a 180 degrees clockwise rotation around the origin, we multiply the coordinates of point A by the matrix:
[cos(180), -sin(180)]
[sin(180), cos(180)]
The cosine of 180 degrees is -1, and the sine of 180 degrees is 0. Applying the matrix multiplication:
A'(x, y) = [5(-1) + 7(0), 5(0) + 7(-1)]
= [-5, -7]
Therefore, the final position of point A after the translation and rotation is A'(-5, -7).
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