Points A and B are at #(9 ,4 )# and #(1 ,5 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #3 #. If point A is now at point B, what are the coordinates of point C?

Question

Points A and B are at #(9   ,4   )# and #(1    ,5   )#, respectively.  Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #3  #.  If point A is now at point B, what are the coordinates of point C?

Sadie Adams · Accepted Answer

The coordinates of #C=(11/2,-16)#

Point #A=((9),(4))# and point #B=((1),(5))#

The rotation of point #A# counterclockwise by #(3/2pi)# tranforms the point #A# into

#A'=((4),(-9))#

Let point #C=((x),(y))#

The dilatation is

#vec(CB)=3vec(CA')#

#((1),(5))-((x),(y))=3*((4),(-9))-((x),(y)))#

Therefore,

#1-x=3(4-x)#

#1-x=12-3x#

#2x=12-1=11#, #=>#, #x=11/2#

#5-y=3(-9-y)#

#5-y=-27-3y#

#2y=-27-5=-32#, #=>#, #y=-16#

The point #C=(11/2,-16)#

Avery Allen · Answer

undefined To find the coordinates of point C, we first need to rotate point A counterclockwise about the origin by $ \frac{3\pi}{2} $ radians and then dilate it by a factor of 3 about an unknown point C to obtain the coordinates of point B.

The rotation of a point $ (x, y) $ counterclockwise about the origin by an angle $ \theta $ radians can be represented by the following equations:

$$ x' = x \cos \theta - y \sin \theta $$
$$ y' = x \sin \theta + y \cos \theta $$

After rotation, the coordinates of point A become:

$$ x' = 9 \cos \left(\frac{3\pi}{2}\right) - 4 \sin \left(\frac{3\pi}{2}\right) $$
$$ y' = 9 \sin \left(\frac{3\pi}{2}\right) + 4 \cos \left(\frac{3\pi}{2}\right) $$

Solving these equations, we get:

$$ x' = -4 $$
$$ y' = 9 $$

Now, we dilate the coordinates $ (x', y') $ about point C by a factor of 3 to obtain the coordinates of point B. Let the coordinates of point C be $ (x_c, y_c) $.

The dilation equations are:

$$ x_b = x_c + 3(x' - x_c) $$
$$ y_b = y_c + 3(y' - y_c) $$

Substituting the values of $ x' $ and $ y' $, and using the coordinates of point B, we have:

$$ 1 = x_c + 3(-4 - x_c) $$
$$ 5 = y_c + 3(9 - y_c) $$

Solving these equations simultaneously, we find:

$$ x_c = -\frac{11}{2} $$
$$ y_c = \frac{11}{2} $$

Therefore, the coordinates of point C are $ \left(-\frac{11}{2}, \frac{11}{2}\right) $.