Points A and B are at #(5 ,9 )# and #(3 ,2 )#, respectively. Point A is rotated counterclockwise about the origin by #pi/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 #(5   ,9   )# and #(3    ,2   )#, respectively.  Point A is rotated counterclockwise about the origin by #pi/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?

Violet Aldrich · Accepted Answer

#C=(-15,13/2)# #"after a counterclockwise rotation about the origin of "pi/2# #• " a point "(x,y)to(-y,x)# #rArrA(5,9)toA'(-9,5)" where A' is the image of A"# #rArrvec(CB)=color(red)(3)vec(CA')# #rArrulb-ulc=3(ula'-ulc)# #rArrulb-ulc=3ula'-3ulc# #rArr2ulc=3ula'-ulb# #color(white)(rArr2ulc)=3((-9),(5))-((3),(2))# #color(white)(rArr2ulc)=((-27),(15))-((3),(2))=((-30),(13))# #rArrulc=1/2((-30),(13))=((-15),(13/2))# #rArrC=(-15,13/2)#

Parker Allen · Answer

undefined To find the coordinates of point C after rotating point A counterclockwise by $ \frac{\pi}{2} $ about the origin and dilating it by a factor of 3 to reach point B, we first need to find the new coordinates of point A after rotation and dilation. Then, we can use these coordinates to determine point C.

1. Rotation of point A counterclockwise by $ \frac{\pi}{2} $ about the origin:

The rotation of a point $ (x, y) $ counterclockwise by angle $ \theta $ about the origin is given by the following formulas:
    
    $$ x' = x \cos(\theta) - y \sin(\theta) $$
    $$ y' = x \sin(\theta) + y \cos(\theta) $$
    
    Given $ (x, y) = (5, 9) $ and $ \theta = \frac{\pi}{2} $, we have:
    
    $$ x' = 5 \cdot \cos\left(\frac{\pi}{2}\right) - 9 \cdot \sin\left(\frac{\pi}{2}\right) = 0 - 9 \cdot 1 = -9 $$
    $$ y' = 5 \cdot \sin\left(\frac{\pi}{2}\right) + 9 \cdot \cos\left(\frac{\pi}{2}\right) = 5 \cdot 1 + 0 = 5 $$

2. Dilation by a factor of 3 about an unknown point C:

Let the coordinates of point C be $ (x_c, y_c) $. The dilation of point A by a factor of 3 about point C means that the distance from point C to point A is multiplied by 3 to reach point B. Therefore, the coordinates of point B are $ (3 \cdot 3, 2 \cdot 3) = (9, 6) $.

Using the coordinates of point B, we can find the coordinates of point C:
    
    $$ x_c = 9 - (-9) = 18 $$
    $$ y_c = 6 - 5 = 1 $$

Therefore, the coordinates of point C are $ (18, 1) $.