Point A(-5,-40) is rotated 90 degrees counterclockwise about the point (-2,-3). What are the coordinate of A' after the rotation? Answer: A'(-1,-6)
Please explain how they got the answer.
that answer cannot possibly be correct
let C(-2,-3) be the centre of rotation.
then CA = <-5+2, -40+3> , where <...> is a vector
= <-3,-37>
using the rotation matrix
cosØ -sinØ
sinØ cosØ
times <-3,-37>
=
0 -1
1 0
times
-3
-37
= <37, -3>
but we have to "move" this vector to its starting point of (-2,-3)
so A' = (35, -6) <-----------
check
slope CA = 37/3
slope CA' = -3/37 , which is the negative reciprocal of the other.
So they are perpendicular.
CA = √( (-37)^2 + 3^2) = √1378
CA' = √1378
so my answer is correct, since I have shown that CA and CA' have the same length and form a 90° angle.
I just noticed that you must have a typo and point A should have been (-5,-4)
If you follow my steps above you will get A' to be (-1,-6)
Oh ok, it makes sense now. Thank you very much for your help!
To rotate a point counterclockwise about another point, we can follow these steps:
1. Calculate the displacement vector from the center of rotation to the point being rotated. In this case, the center of rotation is (-2, -3), and the point being rotated is A(-5, -40). Let's call the displacement vector "v". To find vector v, subtract the coordinates of the center of rotation from the coordinates of A:
v = A - center of rotation = (-5, -40) - (-2, -3)
= (-5 + 2, -40 + 3)
= (-3, -43)
2. The displacement vector v represents the direction and distance from the center of rotation to the original point. To rotate the point counterclockwise by 90 degrees, we need to find the perpendicular vector to v.
To obtain the perpendicular vector, we swap the x and y coordinates of v and negate the new x coordinate. Let's call this perpendicular vector "p":
p = (-v[1], v[0]) = (-(-3), -43) = (3, -43)
3. Add the perpendicular vector p to the center of rotation to calculate the coordinates of the new point after rotation.
A' = center of rotation + p = (-2, -3) + (3, -43)
= (-2 + 3, -3 - 43)
= (-1, -46)
However, if you meant the rotation to be 90 degrees counterclockwise about the point (-3, -4) instead of (-2, -3), the answer will be different. Please clarify if this is the case.