Where does the rotation take the point (4, 1) if a 90 degree rotation counterclockwise about the origin takes (2, 0) to (0, 2) and (0, 3) to (-3, 0).

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william1941 | College Teacher | (Level 3) Valedictorian

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We are given that there is a counter clockwise rotation about the origin.

This means that the original x axis becomes the new y axis and the original y axis becomes the new x axis. But from the fact that (2,0) is taken to (0,2) and (0,3) is taken to (-3,0), we see that the new x axis now points in the direction of the positive y direction but the new y axis points in the direction of the negative x axis. For any point with original coordinates (a, b) the new coordinates are (-b, a).

Therefore the point (4, 1) is taken to (-1, 4).

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neela | High School Teacher | (Level 3) Valedictorian

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90 degree  about the origin (0,0) takes (2,0) to (0,2)

90 degree about the origin takes (0,3) = (-3, 0)

To find where does the rotation of (4,1) about the origin goes to.

Let A bethe point (4,1). and O is the origin (0 , 0).

Then the slope of OA  =  4/1

Length of OA = sqrt(4^2+1) = sqr17.

When  A is rotated by 90 degree , it moves to a place B in 2nd quadrant B (x,y).

OB = OA = sqrt17.

x^2+y^2 = 17...(1)

Then the slope of B = y/x

Product of the slope of OB andOA = -1 as the angle AOB = 90 degrees.

(y/x)*4 = -1

Or y = -4x.........(2).

Put y = -4x in (1):

x^2+(-4x)^2 = 17

17x^2 = 1

x ^2 =  1.

x = sqrt1.

x = 1 or x= -1.

When x =  1, y = -4x = -4*1 =  -4, gives the position of rotation of 90 degree clockwise direction.

When x = -1 , y = -(-1) = 4. Or

B(x, y) =  B ( -1 , 4) is the position of A(1,4) after 90 degree anti clockwise  rotation.

 

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