Points A and B are at #(2 ,2 )# and #(3 ,7 )#, 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?

Answer 1

#color(maroon)("Coordinates of point C " (-9/2, -1/2)#

#A(2,2), B(3,7), "counterclockwise rotation " #pi/2#, "dilation factor" 3#

New coordinates of A after #(3pi)/2# counterclockwise rotation

#A(2,2) rarr A' (-2,2)#

#vec (BC) = (3) vec(A'C)#

#b - c = (3)a' - (3)c#

#2c = (3)a' - b#

#c = (3/2)a' -(1/2) b#

#C((x),(y)) = (3/2)((-2),(2)) - (1/2) ((3),(7)) = ((-9/2),(-1/2))#

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Answer 2

#C=(-9/2,-1/2)#

#"under a counterclockwise rotation about the origin of "pi/2#
#• " a point "(x,y)to(-y,x)#
#A(2,2)toA'(-2,2)" where A' is the image of A "#
#vec(CB)=color(red)(3)vec(CA')#
#ulb-ulc=3(ula'-ulc)#
#ulb-ulc=3ula'-3ulc#
#2ulc=3ula'-ulb#
#color(white)(2ulc)=3((-2),(2))-((3),(7))#
#color(white)(2ulc)=((-6),(6))-((3),(7))=((-9),(-1))#
#ulc=1/2((-9),(-1))=((-9/2),(-1/2))#
#rArrC=(-9/2,-1/2)#
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Answer 3

To find the coordinates of point C, we need to perform two transformations: rotation counterclockwise by (\frac{\pi}{2}) about the origin and dilation by a factor of 3 about point C.

First, let's perform the rotation counterclockwise by (\frac{\pi}{2}) about the origin. The rotation of a point (x, y) counterclockwise by (\frac{\pi}{2}) about the origin results in the point (-y, x).

So, applying this to point A(2, 2), we get (-2, 2).

Now, we need to dilate this point about point C by a factor of 3. Let's denote the coordinates of point C as (x, y).

The dilation formula is given by: [x' = x + k(x_c - x)] [y' = y + k(y_c - y)]

where (x, y) are the original coordinates, (x', y') are the new coordinates after dilation, (x_c, y_c) are the coordinates of the center of dilation, and k is the scale factor.

Given that the new coordinates after dilation are (3, 7), and the scale factor is 3, we can set up the equations:

[3 = -2 + 3(x_c + 2)] [7 = 2 + 3(y_c - 2)]

Solving these equations simultaneously will give us the coordinates of point C.

From the first equation: [3 = -2 + 3x_c + 6] [3 = 3x_c + 4] [3x_c = -1] [x_c = -\frac{1}{3}]

From the second equation: [7 = 2 + 3y_c - 6] [7 = 3y_c - 4] [3y_c = 11] [y_c = \frac{11}{3}]

So, the coordinates of point C are ((- \frac{1}{3}, \frac{11}{3})).

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Answer 4

The coordinates of point C are (2, 7).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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