Answer:

To determine the new coordinates of the point (4,0) after shifting the origin to (1,0) and rotating 45 degrees counterclockwise, we can follow these steps:

Step 1: Shift the origin to (1,0) by subtracting the coordinates of the new origin from the old point.

New point = (4 - 1, 0 - 0) = (3, 0)

Step 2: Rotate the new point 45 degrees counterclockwise. To do this, we can use a rotation matrix.

The rotation matrix for a counterclockwise rotation of θ degrees is:

[ cos(θ) -sin(θ) ]

[ sin(θ) cos(θ) ]

In this case, θ = 45 degrees.

Using the rotation matrix, we can calculate the new coordinates:

New point = [ cos(45) -sin(45) ] * [ 3 ]

[ sin(45) cos(45) ] [ 0 ]

Calculating the values:

New point = [ √2/2 -√2/2 ] * [ 3 ]

[ √2/2 √2/2 ] [ 0 ]

Multiplying the matrices:

New point = [ (√2/2 * 3) + (-√2/2 * 0) ]

[ (√2/2 * 3) + (√2/2 * 0) ]

Simplifying:

New point = [ (3√2)/2 ]

[ (3√2)/2 ]

Therefore, the new coordinates of the point (4,0) after shifting the origin to (1,0) and rotating 45 degrees counterclockwise are approximately (3√2/2, 3√2/2).

Answer:

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