Point A is at $(-2 ,-4 )$ and point B is at $(-3 ,6 )$ . Point A is rotated $(3pi)/2$ clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?

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Answer 1 · 2018-02-09T12:43:03

Distance between A and B changed by $5.93$ unit.

$A(-2,-4) and B (-3,6)$ . Clockwise rotation of point A is

$theta=(3pi)/2 :.$ Counterclockwise rotation of point A is

$theta=(2pi)-(3pi)/2=pi/2$ .

New coordinates of $A(x',y')$ can be found by the fomula

$x'= xcos theta +ysin theta and y'= y cos theta - x sin theta$

$:.x'= (-2)*cos 90+ (-4)*sin90 = -2 *0 -4*1$

$x' = -4 ; y'= -4 * cos 90+ 2 * sin 90 = -4 * 0+2 * 1= 2$

Therefore,new coordinates of A are $(x',y') = (-4,2)$

Distance between two points $(x_1,y_1) and (x_2,y_2)$ is

$D= sqrt((x_1-x_2)^2+(y_1-y_2)^2)$ . Orginal distance between

points $A(-2,-4) and B (-3,6)$ is

$D_o= sqrt((-2+3)^2+(-4-6)^2)=sqrt 101~~ 10.05$ unit.

New distance between points $A(-4,2) and B (-3,6)$ is

$D_n= sqrt((-4+3)^2+(2-6)^2)=sqrt 17~~ 4.12$ unit.

Distance between A and B changed by $10.05-4.12=5.93$ unit .
[Ans]

Point A is at (-2 ,-4 )(-2 ,-4 ) and point B is at (-3 ,6 )(-3 ,6 ). Point A is rotated (3pi)/2 (3pi)/2 clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?