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Points A and B are at #(2 ,7 )# and #(8 ,6 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #4 #. If point A is now at point B, what are the coordinates of point C?

Answer 1

Savannah Aguilar

#C=(-16/3,-34/3)#

#"under a counterclockwise rotation about the origin of "pi#

#• " a point "(x,y)to(-x,-y)#

#rArrA(2,7)toA'(-2,-7)" where A' is the image of A"#

#rArrvec(CB)=color(red)(4)vec(CA')#

#rArrulb-ulc=4(ula'-ulc)#

#rArrulb-ulc=4ula'-4ulc#

#rArr3ulc=4ula'-ulb#

#color(white)(rArr3ulc)=4((-2),(-7))-((8),(6))#

#color(white)(rArr3ulc)=((-8),(-28))-((8),(6))=((-16),(-34))#

#rArrulc=1/3((-16),(-34))=((-16/3),(-34/3))#

#rArrC=(-16/3,-34/3)#

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Answer 2

Theodore Abad

To find the coordinates of point C after the rotation and dilation, we can follow these steps:

Find the new coordinates of point A after the rotation by π radians counterclockwise about the origin. This can be done using the rotation formula:
New_x = Old_x * cos(θ) - Old_y * sin(θ)
New_y = Old_x * sin(θ) + Old_y * cos(θ)
Substituting the values, we get:
New_x = 2 * cos(π) - 7 * sin(π) = -2
New_y = 2 * sin(π) + 7 * cos(π) = -7
Next, dilate the new coordinates of point A by a factor of 4 about point C. The dilation formula is:
New_x = C_x + Dilation_Factor * (Old_x - C_x)
New_y = C_y + Dilation_Factor * (Old_y - C_y)
Substituting the values, we get:
-2 = C_x + 4 * (2 - C_x)
-7 = C_y + 4 * (7 - C_y)
Solve the system of equations to find the coordinates of point C.
From the first equation: -2 = C_x + 8 - 4C_x => 2C_x = 10 => C_x = 5
From the second equation: -7 = C_y + 28 - 4C_y => 3C_y = 21 => C_y = 7

Therefore, the coordinates of point C are (5, 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.